1325:
769:
1320:{\displaystyle {\begin{aligned}{\begin{bmatrix}1&2&1\\-2&-3&1\\3&5&0\end{bmatrix}}&\xrightarrow {2R_{1}+R_{2}\to R_{2}} {\begin{bmatrix}1&2&1\\0&1&3\\3&5&0\end{bmatrix}}\xrightarrow {-3R_{1}+R_{3}\to R_{3}} {\begin{bmatrix}1&2&1\\0&1&3\\0&-1&-3\end{bmatrix}}\\&\xrightarrow {R_{2}+R_{3}\to R_{3}} \,\,{\begin{bmatrix}1&2&1\\0&1&3\\0&0&0\end{bmatrix}}\xrightarrow {-2R_{2}+R_{1}\to R_{1}} {\begin{bmatrix}1&0&-5\\0&1&3\\0&0&0\end{bmatrix}}~.\end{aligned}}}
6398:
3494:. (The order of a minor is the side-length of the square sub-matrix of which it is the determinant.) Like the decomposition rank characterization, this does not give an efficient way of computing the rank, but it is useful theoretically: a single non-zero minor witnesses a lower bound (namely its order) for the rank of the matrix, which can be useful (for example) to prove that certain operations do not lower the rank of a matrix.
6662:
5114:
1969:
4746:
663:. Row operations do not change the row space (hence do not change the row rank), and, being invertible, map the column space to an isomorphic space (hence do not change the column rank). Once in row echelon form, the rank is clearly the same for both row rank and column rank, and equals the number of
3511:
submatrix with non-zero determinant) shows that the rows and columns of that submatrix are linearly independent, and thus those rows and columns of the full matrix are linearly independent (in the full matrix), so the row and column rank are at least as large as the determinantal rank; however, the
4951:
1359:), which are still more numerically robust than Gaussian elimination. Numerical determination of rank requires a criterion for deciding when a value, such as a singular value from the SVD, should be treated as zero, a practical choice which depends on both the matrix and the application.
1760:
4502:
4887:
4629:
3970:
606:
4376:
4578:
3210:
764:
2184:
502:
407:
5140:. If on the other hand, the ranks of these two matrices are equal, then the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has
1395:
possibly bordered by rows and columns of zeros. Again, this changes neither the row rank nor the column rank. It is immediate that both the row and column ranks of this resulting matrix is the number of its nonzero entries.
279:
4235:
2636:
5800:
4406:
4122:
4046:
5721:
5109:{\displaystyle \operatorname {rank} (A)=\operatorname {rank} ({\overline {A}})=\operatorname {rank} (A^{\mathrm {T} })=\operatorname {rank} (A^{*})=\operatorname {rank} (A^{*}A)=\operatorname {rank} (AA^{*}).}
5148:
is the difference between the number of variables and the rank. In this case (and assuming the system of equations is in the real or complex numbers) the system of equations has infinitely many solutions.
5432:
5233:, and thus matrices all have tensor order 2. More precisely, matrices are tensors of type (1,1), having one row index and one column index, also called covariant order 1 and contravariant order 1; see
3314:
3264:
3121:
3063:
2902:
2844:
413:, so the rank is at least 2, but since the third is a linear combination of the first two (the first column plus the second), the three columns are linearly dependent so the rank must be less than 3.
5548:
3754:
774:
4799:
3892:
1964:{\displaystyle 0=c_{1}A\mathbf {x} _{1}+c_{2}A\mathbf {x} _{2}+\cdots +c_{r}A\mathbf {x} _{r}=A(c_{1}\mathbf {x} _{1}+c_{2}\mathbf {x} _{2}+\cdots +c_{r}\mathbf {x} _{r})=A\mathbf {v} ,}
510:
1372:
The fact that the column and row ranks of any matrix are equal forms is fundamental in linear algebra. Many proofs have been given. One of the most elementary ones has been sketched in
4301:
4509:
3359:
5625:
677:
319:
2088:
419:
4804:
2450:
332:
4924:
3471:
5319:
3528:
of the vectors): the equivalence implies that a subset of the rows and a subset of the columns simultaneously define an invertible submatrix (equivalently, if the span of
624:, the statement that the column rank of a matrix equals its row rank is equivalent to the statement that the rank of a matrix is equal to the rank of its transpose, i.e.,
2961:
2931:
3606:
2491:
5550:
is well-defined and injective. We thus obtain the inequality in terms of dimensions of kernel, which can then be converted to the inequality in terms of ranks by the
3141:
222:
299:
213:
4741:{\displaystyle \operatorname {rank} (A^{\mathrm {T} }A)=\operatorname {rank} (AA^{\mathrm {T} })=\operatorname {rank} (A)=\operatorname {rank} (A^{\mathrm {T} }).}
2994:
2743:
1391:
of a matrix has the same row rank and the same column rank as the original matrix. Further elementary column operations allow putting the matrix in the form of an
3410:. Unfortunately, this definition does not suggest an efficient manner to compute the rank (for which it is better to use one of the alternative definitions). See
5648:
5741:
5668:
5572:
2531:
2511:
2401:
4165:
3364:
As in the case of the "dimension of image" characterization, this can be generalized to a definition of the rank of any linear map: the rank of a linear map
180:
if its rank equals the largest possible for a matrix of the same dimensions, which is the lesser of the number of rows and columns. A matrix is said to be
5198:
generalizes to arbitrary tensors; for tensors of order greater than 2 (matrices are order 2 tensors), rank is very hard to compute, unlike for matrices.
4071:
3995:
6256:
5187:, where column rank, row rank, dimension of column space, and dimension of row space of a matrix may be different from the others or may not exist.
5345:
5175:, the rank of the communication matrix of a function gives bounds on the amount of communication needed for two parties to compute the function.
3512:
converse is less straightforward. The equivalence of determinantal rank and column rank is a strengthening of the statement that if the span of
6647:
2576:
5445:
6212:
6129:
6103:
6051:
6011:
3703:
161:
A fundamental result in linear algebra is that the column rank and the row rank are always equal. (Three proofs of this result are given in
5746:
5673:
6174:
6147:
6069:
6039:
5987:
6186:
6159:
6081:
5258:
6178:
6151:
6073:
6043:
6637:
5244:
necessary to express the matrix as a linear combination, and that this definition does agree with matrix rank as here discussed.
6599:
6535:
3269:
3219:
3076:
3018:
2857:
2799:
1356:
504:
has rank 1: there are nonzero columns, so the rank is positive, but any pair of columns is linearly dependent. Similarly, the
4497:{\displaystyle \operatorname {rank} (AB)+\operatorname {rank} (BC)\leq \operatorname {rank} (B)+\operatorname {rank} (ABC).}
5129:
6121:
5234:
6377:
6249:
5263:
5202:
3625:
3621:
3431:
3427:
1348:
65:. This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the "
6482:
6332:
4761:
2643:
2409:
81:. There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics.
6387:
6281:
5214:
5125:
131:
70:
2533:. This definition has the advantage that it can be applied to any linear map without need for a specific matrix.
6627:
6276:
4382:
3557:
660:
6502:
6222:
5551:
5339:
4604:
3582:
rank 1 matrices, where a matrix is defined to have rank 1 if and only if it can be written as a nonzero product
2558:
1347:) can be unreliable, and a rank-revealing decomposition should be used instead. An effective alternative is the
6619:
6225:
5172:
1388:
1380:
46:
5124:
One useful application of calculating the rank of a matrix is the computation of the number of solutions of a
2455:
6686:
6665:
6372:
6242:
3326:
1617:, this establishes the reverse inequality and we obtain the equality of the row rank and the column rank of
1578:
6429:
6362:
6352:
5577:
4882:{\displaystyle 0=\mathbf {x} ^{\mathrm {T} }A^{\mathrm {T} }A\mathbf {x} =\left|A\mathbf {x} \right|^{2}.}
4265:
3487:
1689:
304:
6589:
6444:
6434:
6367:
6312:
74:
165:, below.) This number (i.e., the number of linearly independent rows or columns) is simply called the
115:
In this section, we give some definitions of the rank of a matrix. Many definitions are possible; see
3965:{\displaystyle \operatorname {rank} (AB)\leq \min(\operatorname {rank} (A),\operatorname {rank} (B)).}
1734:. To see why, consider a linear homogeneous relation involving these vectors with scalar coefficients
6454:
6419:
6406:
6297:
5944:
5868:
4902:
3437:
1731:
1384:
650:
58:
39:
601:{\displaystyle A^{\mathrm {T} }={\begin{bmatrix}1&-1\\1&-1\\0&0\\2&-2\end{bmatrix}}}
6632:
6512:
6487:
6337:
3816:
2369:
1415:; it is based upon Mackiw (1995). Both proofs can be found in the book by Banerjee and Roy (2014).
1404:
216:
5295:
4371:{\displaystyle \operatorname {rank} (A)+\operatorname {rank} (B)-n\leq \operatorname {rank} (AB).}
188:
of a matrix is the difference between the lesser of the number of rows and columns, and the rank.
6342:
6113:
5961:
5184:
5137:
3790:
3411:
3001:
1622:
1400:
1327:
The final matrix (in reduced row echelon form) has two non-zero rows and thus the rank of matrix
6230:
4573:{\displaystyle \operatorname {rank} (A+B)\leq \operatorname {rank} (A)+\operatorname {rank} (B)}
3205:{\displaystyle (1)\Leftrightarrow (2)\Leftrightarrow (3)\Leftrightarrow (4)\Leftrightarrow (5)}
2940:
2910:
655:
A common approach to finding the rank of a matrix is to reduce it to a simpler form, generally
6540:
6497:
6424:
6317:
6208:
6182:
6155:
6125:
6099:
6077:
6047:
6007:
5983:
5273:
5268:
4941:
3857:
3585:
1379:
It is straightforward to show that neither the row rank nor the column rank are changed by an
410:
5627:; apply this inequality to the subspace defined by the orthogonal complement of the image of
759:{\displaystyle A={\begin{bmatrix}1&2&1\\-2&-3&1\\3&5&0\end{bmatrix}}}
6545:
6449:
6302:
5953:
5877:
5133:
3524:
of those vectors span the space (equivalently, that one can choose a spanning set that is a
2550:
2179:{\displaystyle c_{1}\mathbf {x} _{1}+c_{2}\mathbf {x} _{2}+\cdots +c_{r}\mathbf {x} _{r}=0.}
1352:
1344:
656:
497:{\displaystyle A={\begin{bmatrix}1&1&0&2\\-1&-1&0&-2\end{bmatrix}}}
284:
198:
6604:
6397:
6357:
6347:
5210:
5161:
4896:
4749:
4600:
4258:
2970:
2719:
1392:
66:
5229:, which is called tensor rank. Tensor order is the number of indices required to write a
766:
can be put in reduced row-echelon form by using the following elementary row operations:
402:{\displaystyle {\begin{bmatrix}1&0&1\\0&1&1\\0&1&1\end{bmatrix}}}
5630:
6609:
6594:
6530:
6265:
5726:
5653:
5557:
5153:
2516:
2496:
2404:
2386:
1340:
31:
5183:
There are different generalizations of the concept of rank to matrices over arbitrary
6680:
6642:
6565:
6525:
6492:
6472:
6031:
5965:
5241:
5165:
5157:
1408:
664:
6221:
Kaw, Autar K. Two
Chapters from the book Introduction to Matrix Algebra: 1. Vectors
5866:
Mackiw, G. (1995), "A Note on the
Equality of the Column and Row Rank of a Matrix",
1597:. This result can be applied to any matrix, so apply the result to the transpose of
1399:
We present two other proofs of this result. The first uses only basic properties of
6575:
6464:
6414:
6307:
6139:
6091:
5957:
5881:
5253:
5226:
4615:
2647:
1412:
135:
50:
6555:
6520:
6477:
6322:
6061:
5195:
4623:
3780:
3686:
3617:
3561:
1648:
1545:
is the matrix which contains the multiples for the bases of the column space of
54:
274:{\displaystyle \operatorname {rank} (\Phi ):=\dim(\operatorname {img} (\Phi ))}
6584:
6327:
5206:
2055:
192:
6382:
3430:, which is the same as the number of non-zero diagonal elements in Σ in the
2689:
617:
505:
151:
1351:(SVD), but there are other less computationally expensive choices, such as
17:
6550:
4230:{\displaystyle XAY={\begin{bmatrix}I_{r}&0\\0&0\\\end{bmatrix}},}
2631:{\displaystyle \mathbf {c} _{1},\mathbf {c} _{2},\dots ,\mathbf {c} _{k}}
5191:
4603:
of the matrix equals the number of columns of the matrix. (This is the
3689:
6560:
5982:, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC,
5230:
2346:
to get the reverse inequality and conclude as in the previous proof.
5795:{\displaystyle \operatorname {rank} (AB)-\operatorname {rank} (ABC)}
1196:
1088:
963:
854:
4117:{\displaystyle \operatorname {rank} (CA)=\operatorname {rank} (A).}
4041:{\displaystyle \operatorname {rank} (AB)=\operatorname {rank} (A).}
5716:{\displaystyle \operatorname {rank} (B)-\operatorname {rank} (BC)}
6234:
5240:
The tensor rank of a matrix can also mean the minimum number of
2561:
states that this definition is equivalent to the preceding one.
6238:
2062:
and, hence, is orthogonal to every vector in the row space of
5427:{\displaystyle \dim \ker(AB)\leq \dim \ker(A)+\dim \ker(B).}
57:) by its columns. This corresponds to the maximal number of
5942:
Wardlaw, William P. (2005), "Row Rank Equals Column Rank",
5156:, the rank of a matrix can be used to determine whether a
1407:. The proof is based upon Wardlaw (2005). The second uses
3309:{\displaystyle \mathbf {c} _{1},\ldots ,\mathbf {c} _{k}}
3259:{\displaystyle \mathbf {c} _{1},\ldots ,\mathbf {c} _{k}}
3116:{\displaystyle \mathbf {r} _{1},\ldots ,\mathbf {r} _{k}}
3058:{\displaystyle \mathbf {r} _{1},\ldots ,\mathbf {r} _{k}}
2897:{\displaystyle \mathbf {c} _{1},\ldots ,\mathbf {c} _{k}}
2839:{\displaystyle \mathbf {c} _{1},\ldots ,\mathbf {c} _{k}}
667:(or basic columns) and also the number of non-zero rows.
5543:{\displaystyle C:\ker(ABC)/\ker(BC)\to \ker(AB)/\ker(B)}
4752:. The null space of the Gram matrix is given by vectors
3478:
Determinantal rank – size of largest non-vanishing minor
6171:
Linear
Algebra: An Introduction to Abstract Mathematics
3749:{\displaystyle \operatorname {rank} (A)\leq \min(m,n).}
2024:
is a linear combination of vectors in the row space of
1343:
computations on computers, basic
Gaussian elimination (
162:
4186:
2684:
is the maximal number of linearly independent rows of
2573:
is the maximal number of linearly independent columns
1246:
1134:
1013:
901:
782:
692:
534:
434:
341:
5749:
5729:
5676:
5656:
5633:
5580:
5560:
5448:
5348:
5298:
4954:
4905:
4807:
4764:
4632:
4588:
are of the same dimension. As a consequence, a rank-
4512:
4409:
4304:
4168:
4074:
3998:
3895:
3706:
3588:
3548:
coordinates on which they are linearly independent).
3440:
3329:
3272:
3222:
3144:
3079:
3021:
2973:
2943:
2913:
2860:
2802:
2722:
2579:
2519:
2499:
2458:
2412:
2389:
2091:
1763:
1525:
th column is formed from the coefficients giving the
772:
680:
513:
422:
335:
307:
287:
225:
201:
5905:
5322:
2664:, which is in fact just the image of the linear map
2318:
linearly independent vectors in the column space of
6618:
6574:
6511:
6463:
6405:
6290:
2354:In all the definitions in this section, the matrix
2201:and so are linearly independent. This implies that
100:; sometimes the parentheses are not written, as in
5794:
5735:
5715:
5662:
5642:
5619:
5566:
5542:
5426:
5313:
5108:
4918:
4881:
4793:
4740:
4572:
4496:
4370:
4229:
4116:
4040:
3964:
3748:
3600:
3465:
3353:
3308:
3258:
3204:
3115:
3057:
2988:
2955:
2925:
2896:
2838:
2737:
2630:
2525:
2505:
2485:
2444:
2395:
2178:
1963:
1319:
758:
600:
496:
401:
313:
293:
273:
207:
5980:Linear Algebra and Matrix Analysis for Statistics
2322:and, hence, the dimension of the column space of
612:has rank 1. Indeed, since the column vectors of
5132:, the system is inconsistent if the rank of the
4794:{\displaystyle A^{\mathrm {T} }A\mathbf {x} =0.}
3917:
3725:
3138:Indeed, the following equivalences are obvious:
1485:can be expressed as a linear combination of the
1373:
4748:This can be shown by proving equality of their
3361:that the row rank is equal to the column rank.
1655:. Therefore, the dimension of the row space of
4378:This is a special case of the next inequality.
3552:Tensor rank – minimum number of simple tensors
6250:
6203:Roger A. Horn and Charles R. Johnson (1985).
4801:If this condition is fulfilled, we also have
2342:. Now apply this result to the transpose of
2276:is obviously a vector in the column space of
8:
2066:. The facts (a) and (b) together imply that
1593:is less than or equal to the column rank of
116:
6098:(4th ed.). Orthogonal Publishing L3C.
3390:can be written as the composition of a map
3212:. For example, to prove (3) from (2), take
2197:were chosen as a basis of the row space of
2072:is orthogonal to itself, which proves that
1387:proceeds by elementary row operations, the
6257:
6243:
6235:
4134:if and only if there exists an invertible
5748:
5728:
5675:
5655:
5632:
5579:
5559:
5520:
5476:
5447:
5347:
5297:
5094:
5063:
5038:
5012:
5011:
4982:
4953:
4906:
4904:
4870:
4860:
4843:
4833:
4832:
4821:
4820:
4815:
4806:
4780:
4770:
4769:
4763:
4725:
4724:
4680:
4679:
4647:
4646:
4631:
4511:
4408:
4303:
4193:
4181:
4167:
4073:
3997:
3894:
3705:
3587:
3457:
3439:
3328:
3300:
3295:
3279:
3274:
3271:
3250:
3245:
3229:
3224:
3221:
3143:
3107:
3102:
3086:
3081:
3078:
3049:
3044:
3028:
3023:
3020:
2972:
2942:
2912:
2888:
2883:
2867:
2862:
2859:
2830:
2825:
2809:
2804:
2801:
2721:
2622:
2617:
2601:
2596:
2586:
2581:
2578:
2518:
2498:
2457:
2436:
2423:
2411:
2388:
2164:
2159:
2152:
2133:
2128:
2121:
2108:
2103:
2096:
2090:
1953:
1938:
1933:
1926:
1907:
1902:
1895:
1882:
1877:
1870:
1851:
1846:
1836:
1817:
1812:
1802:
1789:
1784:
1774:
1762:
1601:. Since the row rank of the transpose of
1241:
1233:
1220:
1207:
1129:
1128:
1127:
1119:
1106:
1093:
1008:
1000:
987:
974:
896:
888:
875:
862:
777:
773:
771:
687:
679:
529:
519:
518:
512:
429:
421:
336:
334:
306:
286:
224:
200:
163:§ Proofs that column rank = row rank
27:Dimension of the column space of a matrix
6118:A (Terse) Introduction to Linear Algebra
5978:Banerjee, Sudipto; Roy, Anindya (2014),
5893:
5861:
5859:
5225:Matrix rank should not be confused with
5213:. It is equal to the linear rank of the
2654:(the column space being the subspace of
1609:and the column rank of the transpose of
1573:form a spanning set of the row space of
1561:is given by a linear combination of the
301:is the dimension of a vector space, and
6229:Mike Brookes: Matrix Reference Manual.
5917:
5815:
5285:
2565:Column rank – dimension of column space
6648:Comparison of linear algebra libraries
5929:
5326:
3354:{\displaystyle (1)\Leftrightarrow (5)}
3266:from (2). To prove (2) from (3), take
409:has rank 2: the first two columns are
5837:
5822:
3650:matrix, and we define the linear map
3486:is the largest order of any non-zero
2338:is no larger than the column rank of
1463:be any basis for the column space of
7:
5833:
5831:
5620:{\displaystyle \dim(AM)\leq \dim(M)}
4592:matrix can be written as the sum of
314:{\displaystyle \operatorname {img} }
4626:are equal. Thus, for real matrices
3216:to be the matrix whose columns are
1589:. This proves that the row rank of
1467:. Place these as the columns of an
1411:and is valid for matrices over the
1376:. Here is a variant of this proof:
215:is defined as the dimension of its
184:if it does not have full rank. The
6175:Undergraduate Texts in Mathematics
6148:Undergraduate Texts in Mathematics
6070:Undergraduate Texts in Mathematics
6040:Undergraduate Texts in Mathematics
5906:Katznelson & Katznelson (2008)
5305:
5013:
4834:
4822:
4771:
4726:
4681:
4648:
4622:and the rank of its corresponding
3692:and cannot be greater than either
3450:
2773:matrix. In fact, for all integers
1403:of vectors, and is valid over any
1374:§ Rank from row echelon forms
1363:Proofs that column rank = row rank
520:
262:
235:
202:
25:
6116:; Katznelson, Yonatan R. (2008).
6004:An introduction to linear algebra
5323:Katznelson & Katznelson (2008
5259:Nonnegative rank (linear algebra)
4926:denotes the complex conjugate of
3793:(or "one-to-one") if and only if
3616:. This notion of rank is called
2676:Row rank – dimension of row space
2513:is the dimension of the image of
2018:. We make two observations: (a)
6661:
6660:
6638:Basic Linear Algebra Subprograms
6396:
6066:Finite-Dimensional Vector Spaces
5136:is greater than the rank of the
4861:
4844:
4816:
4781:
3540:of these vectors span the space
3418:Rank in terms of singular values
3323:It follows from the equivalence
3296:
3275:
3246:
3225:
3103:
3082:
3045:
3024:
2884:
2863:
2826:
2805:
2777:, the following are equivalent:
2618:
2597:
2582:
2445:{\displaystyle f:F^{n}\to F^{m}}
2160:
2129:
2104:
1954:
1934:
1903:
1878:
1847:
1813:
1785:
84:The rank is commonly denoted by
6536:Seven-dimensional cross product
4919:{\displaystyle {\overline {A}}}
4596:rank-1 matrices, but not fewer.
3620:; it can be generalized in the
3466:{\displaystyle A=U\Sigma V^{*}}
2688:; this is the dimension of the
2334:. This proves that row rank of
1553:), which are then used to form
1533:as a linear combination of the
1437:matrix. Let the column rank of
1419:Proof using linear combinations
1357:rank-revealing QR factorization
6207:. Cambridge University Press.
5958:10.1080/0025570X.2005.11953349
5882:10.1080/0025570X.1995.11996337
5789:
5777:
5765:
5756:
5710:
5701:
5689:
5683:
5614:
5608:
5596:
5587:
5537:
5531:
5517:
5508:
5499:
5496:
5487:
5473:
5461:
5418:
5412:
5394:
5388:
5370:
5361:
5308:
5302:
5292:Alternative notation includes
5100:
5084:
5072:
5056:
5044:
5031:
5019:
5004:
4992:
4979:
4967:
4961:
4732:
4717:
4705:
4699:
4687:
4669:
4657:
4639:
4599:The rank of a matrix plus the
4567:
4561:
4549:
4543:
4531:
4519:
4488:
4476:
4464:
4458:
4446:
4437:
4425:
4416:
4362:
4353:
4335:
4329:
4317:
4311:
4108:
4102:
4090:
4081:
4032:
4026:
4014:
4005:
3956:
3953:
3947:
3935:
3929:
3920:
3911:
3902:
3740:
3728:
3719:
3713:
3426:equals the number of non-zero
3348:
3342:
3339:
3336:
3330:
3199:
3193:
3190:
3187:
3181:
3178:
3175:
3169:
3166:
3163:
3157:
3154:
3151:
3145:
2541:Given the same linear mapping
2468:
2462:
2429:
1944:
1863:
1493:. This means that there is an
1226:
1112:
993:
881:
640:Computing the rank of a matrix
268:
265:
259:
250:
238:
232:
1:
6122:American Mathematical Society
5896:p. 200, ch. 3, Definition 2.1
5235:Tensor (intrinsic definition)
2330:) must be at least as big as
1557:as a whole. Now, each row of
6378:Eigenvalues and eigenvectors
6169:Valenza, Robert J. (1993) .
5825:pp. 111-112, §§ 3.115, 3.119
5314:{\displaystyle \rho (\Phi )}
5264:Rank (differential topology)
4987:
4911:
3626:singular value decomposition
3432:singular value decomposition
2660:generated by the columns of
2034:belongs to the row space of
1696:. We claim that the vectors
1349:singular value decomposition
4936:the conjugate transpose of
3827:(in this case, we say that
3819:(or "onto") if and only if
3801:(in this case, we say that
3772:; otherwise, the matrix is
3578:can be written as a sum of
3073:is a linear combination of
2854:is a linear combination of
2549:minus the dimension of the
1647:matrix with entries in the
645:Rank from row echelon forms
616:are the row vectors of the
6703:
5574:is a linear subspace then
5325:, p. 52, §2.5.1) and
5126:system of linear equations
3842:is a square matrix (i.e.,
3555:
2850:such that every column of
2326:(i.e., the column rank of
2259:are linearly independent.
648:
71:system of linear equations
6656:
6394:
6272:
6036:Linear Algebra Done Right
3558:Tensor rank decomposition
3382:of an intermediate space
3378:is the minimal dimension
3130:is less than or equal to
2956:{\displaystyle k\times n}
2926:{\displaystyle m\times k}
2785:is less than or equal to
2403:, there is an associated
2368:matrix over an arbitrary
2079:or, by the definition of
1629:Proof using orthogonality
1569:. Therefore, the rows of
1368:Proof using row reduction
1355:with pivoting (so-called
661:elementary row operations
176:A matrix is said to have
6224:and System of Equations
5853:, ch. II, §10.12, p. 359
5190:Thinking of matrices as
5173:communication complexity
3601:{\displaystyle c\cdot r}
2708:is the smallest integer
2537:Rank in terms of nullity
2486:{\displaystyle f(x)=Ax.}
1389:reduced row echelon form
1381:elementary row operation
670:For example, the matrix
150:is the dimension of the
6144:Advanced Linear Algebra
6002:Mirsky, Leonid (1955).
3756:A matrix that has rank
3570:is the smallest number
3069:such that every row of
3000:is the rank, this is a
2350:Alternative definitions
2058:to every row vector of
1579:Steinitz exchange lemma
321:is the image of a map.
117:Alternative definitions
6363:Row and column vectors
6006:. Dover Publications.
5796:
5737:
5717:
5664:
5644:
5621:
5568:
5544:
5428:
5315:
5144:free parameters where
5130:Rouché–Capelli theorem
5110:
4920:
4883:
4795:
4742:
4574:
4498:
4381:The inequality due to
4372:
4231:
4118:
4042:
3966:
3750:
3624:interpretation of the
3602:
3532:vectors has dimension
3516:vectors has dimension
3467:
3355:
3310:
3260:
3206:
3117:
3059:
2990:
2957:
2927:
2898:
2840:
2739:
2632:
2545:as above, the rank is
2527:
2507:
2487:
2446:
2397:
2180:
1965:
1605:is the column rank of
1321:
760:
602:
498:
403:
315:
295:
275:
209:
119:for several of these.
6368:Row and column spaces
6313:Scalar multiplication
5797:
5738:
5718:
5670:, whose dimension is
5665:
5645:
5622:
5569:
5545:
5429:
5316:
5201:There is a notion of
5111:
4921:
4895:is a matrix over the
4884:
4796:
4743:
4614:is a matrix over the
4575:
4499:
4373:
4232:
4119:
4043:
3967:
3751:
3603:
3468:
3356:
3316:to be the columns of
3311:
3261:
3207:
3118:
3060:
2991:
2958:
2928:
2899:
2841:
2740:
2633:
2528:
2508:
2488:
2447:
2398:
2181:
2028:, which implies that
1966:
1322:
761:
603:
499:
404:
316:
296:
294:{\displaystyle \dim }
276:
210:
208:{\displaystyle \Phi }
75:linear transformation
6503:Gram–Schmidt process
6455:Gaussian elimination
6062:Halmos, Paul Richard
5945:Mathematics Magazine
5869:Mathematics Magazine
5747:
5727:
5674:
5654:
5631:
5578:
5558:
5554:. Alternatively, if
5552:rank–nullity theorem
5446:
5346:
5340:rank–nullity theorem
5329:, p. 90, § 50).
5296:
4952:
4903:
4805:
4762:
4630:
4605:rank–nullity theorem
4510:
4407:
4302:
4166:
4072:
3996:
3893:
3704:
3586:
3438:
3327:
3270:
3220:
3142:
3077:
3019:
2989:{\displaystyle A=CR}
2971:
2941:
2911:
2858:
2800:
2738:{\displaystyle A=CR}
2720:
2577:
2559:rank–nullity theorem
2517:
2497:
2456:
2410:
2387:
2186:But recall that the
2089:
1761:
1732:linearly independent
1692:of the row space of
1521:is the matrix whose
1385:Gaussian elimination
770:
678:
651:Gaussian elimination
511:
420:
411:linearly independent
333:
305:
285:
223:
199:
59:linearly independent
6633:Numerical stability
6513:Multilinear algebra
6488:Inner product space
6338:Linear independence
6114:Katznelson, Yitzhak
5221:Matrices as tensors
5128:. According to the
3608:of a column vector
2781:the column rank of
2716:can be factored as
2225:. It follows that
1613:is the row rank of
1401:linear combinations
1239:
1125:
1006:
894:
6343:Linear combination
5792:
5733:
5723:; its image under
5713:
5660:
5643:{\displaystyle BC}
5640:
5617:
5564:
5540:
5424:
5342:to the inequality
5311:
5138:coefficient matrix
5106:
4916:
4879:
4791:
4738:
4570:
4494:
4403:are defined, then
4368:
4268:’s rank inequality
4227:
4218:
4148:and an invertible
4114:
4038:
3962:
3746:
3598:
3544:there is a set of
3463:
3412:rank factorization
3351:
3306:
3256:
3202:
3113:
3055:
3002:rank factorization
2986:
2953:
2923:
2894:
2836:
2735:
2700:Decomposition rank
2628:
2523:
2503:
2483:
2442:
2393:
2379:Dimension of image
2358:is taken to be an
2176:
1961:
1651:whose row rank is
1623:Rank factorization
1581:, the row rank of
1541:. In other words,
1481:. Every column of
1317:
1315:
1301:
1186:
1071:
953:
840:
756:
750:
598:
592:
494:
488:
399:
393:
311:
291:
271:
205:
6674:
6673:
6541:Geometric algebra
6498:Kronecker product
6333:Linear projection
6318:Vector projection
6214:978-0-521-38632-6
6131:978-0-8218-4419-9
6105:978-1-944325-11-4
6053:978-3-319-11079-0
6013:978-0-486-66434-7
5736:{\displaystyle A}
5663:{\displaystyle B}
5567:{\displaystyle M}
5338:Proof: Apply the
5274:Linear dependence
5269:Multicollinearity
4990:
4914:
4618:then the rank of
3612:and a row vector
2526:{\displaystyle f}
2506:{\displaystyle A}
2396:{\displaystyle A}
2383:Given the matrix
1309:
1240:
1126:
1007:
895:
67:nondegenerateness
16:(Redirected from
6694:
6664:
6663:
6546:Exterior algebra
6483:Hadamard product
6400:
6388:Linear equations
6259:
6252:
6245:
6236:
6218:
6192:
6177:(3rd ed.).
6165:
6150:(2nd ed.).
6135:
6109:
6087:
6072:(2nd ed.).
6057:
6042:(3rd ed.).
6018:
6017:
5999:
5993:
5992:
5975:
5969:
5968:
5939:
5933:
5927:
5921:
5915:
5909:
5903:
5897:
5891:
5885:
5884:
5863:
5854:
5847:
5841:
5835:
5826:
5820:
5803:
5801:
5799:
5798:
5793:
5742:
5740:
5739:
5734:
5722:
5720:
5719:
5714:
5669:
5667:
5666:
5661:
5650:in the image of
5649:
5647:
5646:
5641:
5626:
5624:
5623:
5618:
5573:
5571:
5570:
5565:
5549:
5547:
5546:
5541:
5524:
5480:
5440:
5434:
5433:
5431:
5430:
5425:
5336:
5330:
5320:
5318:
5317:
5312:
5290:
5211:smooth manifolds
5171:In the field of
5147:
5143:
5134:augmented matrix
5115:
5113:
5112:
5107:
5099:
5098:
5068:
5067:
5043:
5042:
5018:
5017:
5016:
4991:
4983:
4947:
4939:
4935:
4929:
4925:
4923:
4922:
4917:
4915:
4907:
4894:
4888:
4886:
4885:
4880:
4875:
4874:
4869:
4865:
4864:
4847:
4839:
4838:
4837:
4827:
4826:
4825:
4819:
4800:
4798:
4797:
4792:
4784:
4776:
4775:
4774:
4757:
4747:
4745:
4744:
4739:
4731:
4730:
4729:
4686:
4685:
4684:
4653:
4652:
4651:
4621:
4613:
4595:
4591:
4587:
4583:
4579:
4577:
4576:
4571:
4503:
4501:
4500:
4495:
4402:
4396:
4390:
4377:
4375:
4374:
4369:
4297:
4287:
4283:
4273:
4257:
4247:
4236:
4234:
4233:
4228:
4223:
4222:
4198:
4197:
4161:
4157:
4147:
4143:
4133:
4129:
4123:
4121:
4120:
4115:
4067:
4063:
4053:
4047:
4045:
4044:
4039:
3991:
3987:
3977:
3971:
3969:
3968:
3963:
3888:
3878:
3871:
3867:
3863:
3855:
3851:
3841:
3830:
3826:
3822:
3814:
3807:full column rank
3804:
3800:
3796:
3788:
3768:is said to have
3767:
3755:
3753:
3752:
3747:
3699:
3695:
3684:
3670:
3653:
3649:
3639:
3622:separable models
3615:
3611:
3607:
3605:
3604:
3599:
3581:
3577:
3573:
3569:
3547:
3539:
3535:
3531:
3523:
3519:
3515:
3510:
3500:
3497:A non-vanishing
3493:
3485:
3474:
3472:
3470:
3469:
3464:
3462:
3461:
3425:
3409:
3399:
3389:
3385:
3381:
3377:
3360:
3358:
3357:
3352:
3319:
3315:
3313:
3312:
3307:
3305:
3304:
3299:
3284:
3283:
3278:
3265:
3263:
3262:
3257:
3255:
3254:
3249:
3234:
3233:
3228:
3215:
3211:
3209:
3208:
3203:
3133:
3129:
3126:the row rank of
3122:
3120:
3119:
3114:
3112:
3111:
3106:
3091:
3090:
3085:
3072:
3068:
3064:
3062:
3061:
3056:
3054:
3053:
3048:
3033:
3032:
3027:
3014:
3007:
2999:
2995:
2993:
2992:
2987:
2966:
2962:
2960:
2959:
2954:
2936:
2932:
2930:
2929:
2924:
2903:
2901:
2900:
2895:
2893:
2892:
2887:
2872:
2871:
2866:
2853:
2849:
2845:
2843:
2842:
2837:
2835:
2834:
2829:
2814:
2813:
2808:
2795:
2788:
2784:
2776:
2772:
2762:
2758:
2748:
2744:
2742:
2741:
2736:
2715:
2711:
2707:
2695:
2687:
2683:
2671:
2667:
2663:
2659:
2653:
2641:
2637:
2635:
2634:
2629:
2627:
2626:
2621:
2606:
2605:
2600:
2591:
2590:
2585:
2572:
2556:
2548:
2544:
2532:
2530:
2529:
2524:
2512:
2510:
2509:
2504:
2492:
2490:
2489:
2484:
2451:
2449:
2448:
2443:
2441:
2440:
2428:
2427:
2402:
2400:
2399:
2394:
2374:
2367:
2357:
2345:
2341:
2337:
2333:
2329:
2325:
2321:
2317:
2313:
2279:
2275:
2258:
2224:
2200:
2196:
2185:
2183:
2182:
2177:
2169:
2168:
2163:
2157:
2156:
2138:
2137:
2132:
2126:
2125:
2113:
2112:
2107:
2101:
2100:
2084:
2078:
2071:
2065:
2061:
2053:
2047:
2038:, and (b) since
2037:
2033:
2027:
2023:
2017:
1970:
1968:
1967:
1962:
1957:
1943:
1942:
1937:
1931:
1930:
1912:
1911:
1906:
1900:
1899:
1887:
1886:
1881:
1875:
1874:
1856:
1855:
1850:
1841:
1840:
1822:
1821:
1816:
1807:
1806:
1794:
1793:
1788:
1779:
1778:
1756:
1729:
1695:
1687:
1662:
1658:
1654:
1646:
1636:
1620:
1616:
1612:
1608:
1604:
1600:
1596:
1592:
1588:
1584:
1576:
1572:
1568:
1564:
1560:
1556:
1552:
1548:
1544:
1540:
1536:
1532:
1528:
1524:
1520:
1516:
1506:
1502:
1492:
1488:
1484:
1480:
1476:
1466:
1462:
1444:
1440:
1436:
1426:
1353:QR decomposition
1345:LU decomposition
1339:When applied to
1330:
1326:
1324:
1323:
1318:
1316:
1307:
1306:
1305:
1238:
1237:
1225:
1224:
1212:
1211:
1192:
1191:
1190:
1124:
1123:
1111:
1110:
1098:
1097:
1084:
1080:
1076:
1075:
1005:
1004:
992:
991:
979:
978:
959:
958:
957:
893:
892:
880:
879:
867:
866:
850:
845:
844:
765:
763:
762:
757:
755:
754:
673:
657:row echelon form
635:
623:
615:
611:
607:
605:
604:
599:
597:
596:
525:
524:
523:
503:
501:
500:
495:
493:
492:
408:
406:
405:
400:
398:
397:
320:
318:
317:
312:
300:
298:
297:
292:
280:
278:
277:
272:
214:
212:
211:
206:
172:
157:
149:
141:
129:
111:Main definitions
106:
99:
91:
80:
64:
44:
21:
6702:
6701:
6697:
6696:
6695:
6693:
6692:
6691:
6677:
6676:
6675:
6670:
6652:
6614:
6570:
6507:
6459:
6401:
6392:
6358:Change of basis
6348:Multilinear map
6286:
6268:
6263:
6215:
6205:Matrix Analysis
6202:
6199:
6197:Further reading
6189:
6168:
6162:
6138:
6132:
6112:
6106:
6090:
6084:
6060:
6054:
6030:
6027:
6022:
6021:
6014:
6001:
6000:
5996:
5990:
5977:
5976:
5972:
5941:
5940:
5936:
5928:
5924:
5916:
5912:
5904:
5900:
5894:Hefferon (2020)
5892:
5888:
5865:
5864:
5857:
5848:
5844:
5836:
5829:
5821:
5817:
5812:
5807:
5806:
5745:
5744:
5725:
5724:
5672:
5671:
5652:
5651:
5629:
5628:
5576:
5575:
5556:
5555:
5444:
5443:
5441:
5437:
5344:
5343:
5337:
5333:
5294:
5293:
5291:
5287:
5282:
5250:
5223:
5181:
5145:
5141:
5122:
5090:
5059:
5034:
5007:
4950:
4949:
4945:
4937:
4931:
4927:
4901:
4900:
4897:complex numbers
4892:
4856:
4852:
4851:
4828:
4814:
4803:
4802:
4765:
4760:
4759:
4753:
4720:
4675:
4642:
4628:
4627:
4619:
4611:
4593:
4589:
4585:
4581:
4508:
4507:
4506:Subadditivity:
4405:
4404:
4398:
4392:
4386:
4300:
4299:
4289:
4285:
4275:
4271:
4259:identity matrix
4249:
4246:
4238:
4217:
4216:
4211:
4205:
4204:
4199:
4189:
4182:
4164:
4163:
4159:
4149:
4145:
4135:
4131:
4127:
4070:
4069:
4065:
4064:matrix of rank
4055:
4051:
3994:
3993:
3989:
3988:matrix of rank
3979:
3975:
3891:
3890:
3880:
3876:
3872:has full rank).
3869:
3865:
3861:
3860:if and only if
3853:
3843:
3839:
3828:
3824:
3820:
3812:
3802:
3798:
3794:
3786:
3757:
3702:
3701:
3697:
3693:
3676:
3675:The rank of an
3655:
3651:
3641:
3637:
3636:We assume that
3634:
3613:
3609:
3584:
3583:
3579:
3575:
3571:
3567:
3564:
3556:Main articles:
3554:
3545:
3537:
3533:
3529:
3521:
3517:
3513:
3502:
3498:
3491:
3483:
3480:
3453:
3436:
3435:
3434:
3428:singular values
3423:
3420:
3401:
3391:
3387:
3383:
3379:
3365:
3325:
3324:
3317:
3294:
3273:
3268:
3267:
3244:
3223:
3218:
3217:
3213:
3140:
3139:
3131:
3127:
3101:
3080:
3075:
3074:
3070:
3066:
3043:
3022:
3017:
3016:
3012:
3005:
2997:
2969:
2968:
2964:
2939:
2938:
2934:
2909:
2908:
2907:there exist an
2882:
2861:
2856:
2855:
2851:
2847:
2824:
2803:
2798:
2797:
2793:
2786:
2782:
2774:
2764:
2760:
2750:
2746:
2718:
2717:
2713:
2709:
2705:
2702:
2693:
2685:
2681:
2678:
2669:
2665:
2661:
2655:
2651:
2639:
2616:
2595:
2580:
2575:
2574:
2570:
2567:
2554:
2546:
2542:
2539:
2515:
2514:
2495:
2494:
2454:
2453:
2432:
2419:
2408:
2407:
2385:
2384:
2381:
2372:
2359:
2355:
2352:
2343:
2339:
2335:
2331:
2327:
2323:
2319:
2315:
2312:
2300:
2290:
2281:
2277:
2274:
2263:
2257:
2245:
2235:
2226:
2221:
2215:
2208:
2202:
2198:
2195:
2187:
2158:
2148:
2127:
2117:
2102:
2092:
2087:
2086:
2080:
2073:
2067:
2063:
2059:
2049:
2039:
2035:
2029:
2025:
2019:
2016:
2007:
2001:
1995:
1988:
1982:
1972:
1932:
1922:
1901:
1891:
1876:
1866:
1845:
1832:
1811:
1798:
1783:
1770:
1759:
1758:
1754:
1748:
1741:
1735:
1728:
1716:
1706:
1697:
1693:
1686:
1677:
1670:
1664:
1660:
1656:
1652:
1642: ×
1638:
1634:
1631:
1618:
1614:
1610:
1606:
1602:
1598:
1594:
1590:
1586:
1582:
1574:
1570:
1566:
1562:
1558:
1554:
1550:
1546:
1542:
1538:
1534:
1530:
1526:
1522:
1518:
1508:
1504:
1494:
1490:
1486:
1482:
1478:
1468:
1464:
1461:
1452:
1446:
1442:
1438:
1428:
1424:
1421:
1393:identity matrix
1370:
1365:
1337:
1328:
1314:
1313:
1300:
1299:
1294:
1289:
1283:
1282:
1277:
1272:
1266:
1265:
1257:
1252:
1242:
1229:
1216:
1203:
1185:
1184:
1179:
1174:
1168:
1167:
1162:
1157:
1151:
1150:
1145:
1140:
1130:
1115:
1102:
1089:
1078:
1077:
1070:
1069:
1061:
1053:
1047:
1046:
1041:
1036:
1030:
1029:
1024:
1019:
1009:
996:
983:
970:
952:
951:
946:
941:
935:
934:
929:
924:
918:
917:
912:
907:
897:
884:
871:
858:
846:
839:
838:
833:
828:
822:
821:
816:
808:
799:
798:
793:
788:
778:
768:
767:
749:
748:
743:
738:
732:
731:
726:
718:
709:
708:
703:
698:
688:
676:
675:
671:
653:
647:
642:
625:
621:
613:
609:
591:
590:
582:
576:
575:
570:
564:
563:
555:
549:
548:
540:
530:
514:
509:
508:
487:
486:
478:
473:
465:
456:
455:
450:
445:
440:
430:
418:
417:
392:
391:
386:
381:
375:
374:
369:
364:
358:
357:
352:
347:
337:
331:
330:
327:
303:
302:
283:
282:
221:
220:
197:
196:
186:rank deficiency
170:
155:
147:
139:
127:
113:
101:
93:
85:
78:
62:
42:
28:
23:
22:
15:
12:
11:
5:
6700:
6698:
6690:
6689:
6687:Linear algebra
6679:
6678:
6672:
6671:
6669:
6668:
6657:
6654:
6653:
6651:
6650:
6645:
6640:
6635:
6630:
6628:Floating-point
6624:
6622:
6616:
6615:
6613:
6612:
6610:Tensor product
6607:
6602:
6597:
6595:Function space
6592:
6587:
6581:
6579:
6572:
6571:
6569:
6568:
6563:
6558:
6553:
6548:
6543:
6538:
6533:
6531:Triple product
6528:
6523:
6517:
6515:
6509:
6508:
6506:
6505:
6500:
6495:
6490:
6485:
6480:
6475:
6469:
6467:
6461:
6460:
6458:
6457:
6452:
6447:
6445:Transformation
6442:
6437:
6435:Multiplication
6432:
6427:
6422:
6417:
6411:
6409:
6403:
6402:
6395:
6393:
6391:
6390:
6385:
6380:
6375:
6370:
6365:
6360:
6355:
6350:
6345:
6340:
6335:
6330:
6325:
6320:
6315:
6310:
6305:
6300:
6294:
6292:
6291:Basic concepts
6288:
6287:
6285:
6284:
6279:
6273:
6270:
6269:
6266:Linear algebra
6264:
6262:
6261:
6254:
6247:
6239:
6233:
6232:
6227:
6219:
6213:
6198:
6195:
6194:
6193:
6187:
6166:
6160:
6136:
6130:
6110:
6104:
6096:Linear Algebra
6088:
6082:
6058:
6052:
6032:Axler, Sheldon
6026:
6023:
6020:
6019:
6012:
5994:
5989:978-1420095388
5988:
5970:
5952:(4): 316–318,
5934:
5922:
5918:Valenza (1993)
5910:
5908:p. 52, § 2.5.1
5898:
5886:
5876:(4): 285–286,
5855:
5842:
5827:
5814:
5813:
5811:
5808:
5805:
5804:
5791:
5788:
5785:
5782:
5779:
5776:
5773:
5770:
5767:
5764:
5761:
5758:
5755:
5752:
5743:has dimension
5732:
5712:
5709:
5706:
5703:
5700:
5697:
5694:
5691:
5688:
5685:
5682:
5679:
5659:
5639:
5636:
5616:
5613:
5610:
5607:
5604:
5601:
5598:
5595:
5592:
5589:
5586:
5583:
5563:
5539:
5536:
5533:
5530:
5527:
5523:
5519:
5516:
5513:
5510:
5507:
5504:
5501:
5498:
5495:
5492:
5489:
5486:
5483:
5479:
5475:
5472:
5469:
5466:
5463:
5460:
5457:
5454:
5451:
5442:Proof. The map
5435:
5423:
5420:
5417:
5414:
5411:
5408:
5405:
5402:
5399:
5396:
5393:
5390:
5387:
5384:
5381:
5378:
5375:
5372:
5369:
5366:
5363:
5360:
5357:
5354:
5351:
5331:
5310:
5307:
5304:
5301:
5284:
5283:
5281:
5278:
5277:
5276:
5271:
5266:
5261:
5256:
5249:
5246:
5242:simple tensors
5222:
5219:
5180:
5179:Generalization
5177:
5154:control theory
5121:
5118:
5117:
5116:
5105:
5102:
5097:
5093:
5089:
5086:
5083:
5080:
5077:
5074:
5071:
5066:
5062:
5058:
5055:
5052:
5049:
5046:
5041:
5037:
5033:
5030:
5027:
5024:
5021:
5015:
5010:
5006:
5003:
5000:
4997:
4994:
4989:
4986:
4981:
4978:
4975:
4972:
4969:
4966:
4963:
4960:
4957:
4913:
4910:
4889:
4878:
4873:
4868:
4863:
4859:
4855:
4850:
4846:
4842:
4836:
4831:
4824:
4818:
4813:
4810:
4790:
4787:
4783:
4779:
4773:
4768:
4737:
4734:
4728:
4723:
4719:
4716:
4713:
4710:
4707:
4704:
4701:
4698:
4695:
4692:
4689:
4683:
4678:
4674:
4671:
4668:
4665:
4662:
4659:
4656:
4650:
4645:
4641:
4638:
4635:
4608:
4597:
4569:
4566:
4563:
4560:
4557:
4554:
4551:
4548:
4545:
4542:
4539:
4536:
4533:
4530:
4527:
4524:
4521:
4518:
4515:
4504:
4493:
4490:
4487:
4484:
4481:
4478:
4475:
4472:
4469:
4466:
4463:
4460:
4457:
4454:
4451:
4448:
4445:
4442:
4439:
4436:
4433:
4430:
4427:
4424:
4421:
4418:
4415:
4412:
4379:
4367:
4364:
4361:
4358:
4355:
4352:
4349:
4346:
4343:
4340:
4337:
4334:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4310:
4307:
4262:
4242:
4226:
4221:
4215:
4212:
4210:
4207:
4206:
4203:
4200:
4196:
4192:
4188:
4187:
4185:
4180:
4177:
4174:
4171:
4124:
4113:
4110:
4107:
4104:
4101:
4098:
4095:
4092:
4089:
4086:
4083:
4080:
4077:
4048:
4037:
4034:
4031:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
4007:
4004:
4001:
3972:
3961:
3958:
3955:
3952:
3949:
3946:
3943:
3940:
3937:
3934:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3873:
3836:
3810:
3784:
3783:has rank zero.
3777:
3774:rank deficient
3745:
3742:
3739:
3736:
3733:
3730:
3727:
3724:
3721:
3718:
3715:
3712:
3709:
3633:
3630:
3597:
3594:
3591:
3553:
3550:
3479:
3476:
3460:
3456:
3452:
3449:
3446:
3443:
3419:
3416:
3350:
3347:
3344:
3341:
3338:
3335:
3332:
3303:
3298:
3293:
3290:
3287:
3282:
3277:
3253:
3248:
3243:
3240:
3237:
3232:
3227:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3136:
3135:
3124:
3110:
3105:
3100:
3097:
3094:
3089:
3084:
3052:
3047:
3042:
3039:
3036:
3031:
3026:
3009:
2985:
2982:
2979:
2976:
2952:
2949:
2946:
2922:
2919:
2916:
2905:
2891:
2886:
2881:
2878:
2875:
2870:
2865:
2833:
2828:
2823:
2820:
2817:
2812:
2807:
2790:
2734:
2731:
2728:
2725:
2701:
2698:
2677:
2674:
2668:associated to
2642:; this is the
2625:
2620:
2615:
2612:
2609:
2604:
2599:
2594:
2589:
2584:
2566:
2563:
2538:
2535:
2522:
2502:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2439:
2435:
2431:
2426:
2422:
2418:
2415:
2405:linear mapping
2392:
2380:
2377:
2351:
2348:
2308:
2298:
2288:
2270:
2253:
2243:
2233:
2219:
2213:
2206:
2191:
2175:
2172:
2167:
2162:
2155:
2151:
2147:
2144:
2141:
2136:
2131:
2124:
2120:
2116:
2111:
2106:
2099:
2095:
2012:
2005:
1999:
1993:
1986:
1980:
1960:
1956:
1952:
1949:
1946:
1941:
1936:
1929:
1925:
1921:
1918:
1915:
1910:
1905:
1898:
1894:
1890:
1885:
1880:
1873:
1869:
1865:
1862:
1859:
1854:
1849:
1844:
1839:
1835:
1831:
1828:
1825:
1820:
1815:
1810:
1805:
1801:
1797:
1792:
1787:
1782:
1777:
1773:
1769:
1766:
1752:
1746:
1739:
1724:
1714:
1704:
1682:
1675:
1668:
1630:
1627:
1585:cannot exceed
1457:
1450:
1420:
1417:
1369:
1366:
1364:
1361:
1341:floating point
1336:
1333:
1312:
1304:
1298:
1295:
1293:
1290:
1288:
1285:
1284:
1281:
1278:
1276:
1273:
1271:
1268:
1267:
1264:
1261:
1258:
1256:
1253:
1251:
1248:
1247:
1245:
1236:
1232:
1228:
1223:
1219:
1215:
1210:
1206:
1202:
1199:
1195:
1189:
1183:
1180:
1178:
1175:
1173:
1170:
1169:
1166:
1163:
1161:
1158:
1156:
1153:
1152:
1149:
1146:
1144:
1141:
1139:
1136:
1135:
1133:
1122:
1118:
1114:
1109:
1105:
1101:
1096:
1092:
1087:
1083:
1081:
1079:
1074:
1068:
1065:
1062:
1060:
1057:
1054:
1052:
1049:
1048:
1045:
1042:
1040:
1037:
1035:
1032:
1031:
1028:
1025:
1023:
1020:
1018:
1015:
1014:
1012:
1003:
999:
995:
990:
986:
982:
977:
973:
969:
966:
962:
956:
950:
947:
945:
942:
940:
937:
936:
933:
930:
928:
925:
923:
920:
919:
916:
913:
911:
908:
906:
903:
902:
900:
891:
887:
883:
878:
874:
870:
865:
861:
857:
853:
849:
847:
843:
837:
834:
832:
829:
827:
824:
823:
820:
817:
815:
812:
809:
807:
804:
801:
800:
797:
794:
792:
789:
787:
784:
783:
781:
776:
775:
753:
747:
744:
742:
739:
737:
734:
733:
730:
727:
725:
722:
719:
717:
714:
711:
710:
707:
704:
702:
699:
697:
694:
693:
691:
686:
683:
649:Main article:
646:
643:
641:
638:
595:
589:
586:
583:
581:
578:
577:
574:
571:
569:
566:
565:
562:
559:
556:
554:
551:
550:
547:
544:
541:
539:
536:
535:
533:
528:
522:
517:
491:
485:
482:
479:
477:
474:
472:
469:
466:
464:
461:
458:
457:
454:
451:
449:
446:
444:
441:
439:
436:
435:
433:
428:
425:
396:
390:
387:
385:
382:
380:
377:
376:
373:
370:
368:
365:
363:
360:
359:
356:
353:
351:
348:
346:
343:
342:
340:
326:
323:
310:
290:
270:
267:
264:
261:
258:
255:
252:
249:
246:
243:
240:
237:
234:
231:
228:
204:
191:The rank of a
182:rank-deficient
112:
109:
53:generated (or
32:linear algebra
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6699:
6688:
6685:
6684:
6682:
6667:
6659:
6658:
6655:
6649:
6646:
6644:
6643:Sparse matrix
6641:
6639:
6636:
6634:
6631:
6629:
6626:
6625:
6623:
6621:
6617:
6611:
6608:
6606:
6603:
6601:
6598:
6596:
6593:
6591:
6588:
6586:
6583:
6582:
6580:
6578:constructions
6577:
6573:
6567:
6566:Outermorphism
6564:
6562:
6559:
6557:
6554:
6552:
6549:
6547:
6544:
6542:
6539:
6537:
6534:
6532:
6529:
6527:
6526:Cross product
6524:
6522:
6519:
6518:
6516:
6514:
6510:
6504:
6501:
6499:
6496:
6494:
6493:Outer product
6491:
6489:
6486:
6484:
6481:
6479:
6476:
6474:
6473:Orthogonality
6471:
6470:
6468:
6466:
6462:
6456:
6453:
6451:
6450:Cramer's rule
6448:
6446:
6443:
6441:
6438:
6436:
6433:
6431:
6428:
6426:
6423:
6421:
6420:Decomposition
6418:
6416:
6413:
6412:
6410:
6408:
6404:
6399:
6389:
6386:
6384:
6381:
6379:
6376:
6374:
6371:
6369:
6366:
6364:
6361:
6359:
6356:
6354:
6351:
6349:
6346:
6344:
6341:
6339:
6336:
6334:
6331:
6329:
6326:
6324:
6321:
6319:
6316:
6314:
6311:
6309:
6306:
6304:
6301:
6299:
6296:
6295:
6293:
6289:
6283:
6280:
6278:
6275:
6274:
6271:
6267:
6260:
6255:
6253:
6248:
6246:
6241:
6240:
6237:
6231:
6228:
6226:
6223:
6220:
6216:
6210:
6206:
6201:
6200:
6196:
6190:
6188:3-540-94099-5
6184:
6180:
6176:
6172:
6167:
6163:
6161:0-387-24766-1
6157:
6153:
6149:
6145:
6141:
6140:Roman, Steven
6137:
6133:
6127:
6123:
6119:
6115:
6111:
6107:
6101:
6097:
6093:
6092:Hefferon, Jim
6089:
6085:
6083:0-387-90093-4
6079:
6075:
6071:
6067:
6063:
6059:
6055:
6049:
6045:
6041:
6037:
6033:
6029:
6028:
6024:
6015:
6009:
6005:
5998:
5995:
5991:
5985:
5981:
5974:
5971:
5967:
5963:
5959:
5955:
5951:
5947:
5946:
5938:
5935:
5931:
5930:Halmos (1974)
5926:
5923:
5919:
5914:
5911:
5907:
5902:
5899:
5895:
5890:
5887:
5883:
5879:
5875:
5871:
5870:
5862:
5860:
5856:
5852:
5846:
5843:
5840:p. 48, § 1.16
5839:
5834:
5832:
5828:
5824:
5819:
5816:
5809:
5786:
5783:
5780:
5774:
5771:
5768:
5762:
5759:
5753:
5750:
5730:
5707:
5704:
5698:
5695:
5692:
5686:
5680:
5677:
5657:
5637:
5634:
5611:
5605:
5602:
5599:
5593:
5590:
5584:
5581:
5561:
5553:
5534:
5528:
5525:
5521:
5514:
5511:
5505:
5502:
5493:
5490:
5484:
5481:
5477:
5470:
5467:
5464:
5458:
5455:
5452:
5449:
5439:
5436:
5421:
5415:
5409:
5406:
5403:
5400:
5397:
5391:
5385:
5382:
5379:
5376:
5373:
5367:
5364:
5358:
5355:
5352:
5349:
5341:
5335:
5332:
5328:
5324:
5299:
5289:
5286:
5279:
5275:
5272:
5270:
5267:
5265:
5262:
5260:
5257:
5255:
5252:
5251:
5247:
5245:
5243:
5238:
5237:for details.
5236:
5232:
5228:
5220:
5218:
5216:
5212:
5208:
5204:
5199:
5197:
5193:
5188:
5186:
5178:
5176:
5174:
5169:
5167:
5163:
5159:
5158:linear system
5155:
5150:
5139:
5135:
5131:
5127:
5119:
5103:
5095:
5091:
5087:
5081:
5078:
5075:
5069:
5064:
5060:
5053:
5050:
5047:
5039:
5035:
5028:
5025:
5022:
5008:
5001:
4998:
4995:
4984:
4976:
4973:
4970:
4964:
4958:
4955:
4943:
4934:
4908:
4898:
4890:
4876:
4871:
4866:
4857:
4853:
4848:
4840:
4829:
4811:
4808:
4788:
4785:
4777:
4766:
4756:
4751:
4735:
4721:
4714:
4711:
4708:
4702:
4696:
4693:
4690:
4676:
4672:
4666:
4663:
4660:
4654:
4643:
4636:
4633:
4625:
4617:
4609:
4606:
4602:
4598:
4564:
4558:
4555:
4552:
4546:
4540:
4537:
4534:
4528:
4525:
4522:
4516:
4513:
4505:
4491:
4485:
4482:
4479:
4473:
4470:
4467:
4461:
4455:
4452:
4449:
4443:
4440:
4434:
4431:
4428:
4422:
4419:
4413:
4410:
4401:
4395:
4389:
4384:
4380:
4365:
4359:
4356:
4350:
4347:
4344:
4341:
4338:
4332:
4326:
4323:
4320:
4314:
4308:
4305:
4296:
4292:
4282:
4278:
4269:
4267:
4263:
4260:
4256:
4252:
4245:
4241:
4224:
4219:
4213:
4208:
4201:
4194:
4190:
4183:
4178:
4175:
4172:
4169:
4156:
4152:
4142:
4138:
4125:
4111:
4105:
4099:
4096:
4093:
4087:
4084:
4078:
4075:
4062:
4058:
4049:
4035:
4029:
4023:
4020:
4017:
4011:
4008:
4002:
3999:
3986:
3982:
3973:
3959:
3950:
3944:
3941:
3938:
3932:
3926:
3923:
3914:
3908:
3905:
3899:
3896:
3889:matrix, then
3887:
3883:
3874:
3859:
3850:
3846:
3837:
3834:
3833:full row rank
3818:
3811:
3808:
3792:
3785:
3782:
3778:
3775:
3771:
3765:
3761:
3743:
3737:
3734:
3731:
3722:
3716:
3710:
3707:
3691:
3688:
3683:
3679:
3674:
3673:
3672:
3669:
3666:
3662:
3658:
3648:
3644:
3631:
3629:
3627:
3623:
3619:
3595:
3592:
3589:
3563:
3559:
3551:
3549:
3543:
3527:
3509:
3505:
3495:
3489:
3477:
3475:
3458:
3454:
3447:
3444:
3441:
3433:
3429:
3417:
3415:
3414:for details.
3413:
3408:
3404:
3398:
3394:
3376:
3372:
3368:
3362:
3345:
3333:
3321:
3301:
3291:
3288:
3285:
3280:
3251:
3241:
3238:
3235:
3230:
3196:
3184:
3172:
3160:
3148:
3125:
3108:
3098:
3095:
3092:
3087:
3050:
3040:
3037:
3034:
3029:
3010:
3003:
2983:
2980:
2977:
2974:
2950:
2947:
2944:
2920:
2917:
2914:
2906:
2889:
2879:
2876:
2873:
2868:
2831:
2821:
2818:
2815:
2810:
2791:
2780:
2779:
2778:
2771:
2767:
2757:
2753:
2732:
2729:
2726:
2723:
2699:
2697:
2691:
2675:
2673:
2658:
2649:
2645:
2623:
2613:
2610:
2607:
2602:
2592:
2587:
2564:
2562:
2560:
2552:
2536:
2534:
2520:
2500:
2480:
2477:
2474:
2471:
2465:
2459:
2437:
2433:
2424:
2420:
2416:
2413:
2406:
2390:
2378:
2376:
2371:
2366:
2362:
2349:
2347:
2311:
2307:
2304:
2297:
2294:
2287:
2284:
2273:
2269:
2266:
2260:
2256:
2252:
2249:
2242:
2239:
2232:
2229:
2222:
2212:
2205:
2194:
2190:
2173:
2170:
2165:
2153:
2149:
2145:
2142:
2139:
2134:
2122:
2118:
2114:
2109:
2097:
2093:
2083:
2076:
2070:
2057:
2052:
2048:, the vector
2045:
2042:
2032:
2022:
2015:
2011:
2008:
1998:
1992:
1985:
1979:
1975:
1958:
1950:
1947:
1939:
1927:
1923:
1919:
1916:
1913:
1908:
1896:
1892:
1888:
1883:
1871:
1867:
1860:
1857:
1852:
1842:
1837:
1833:
1829:
1826:
1823:
1818:
1808:
1803:
1799:
1795:
1790:
1780:
1775:
1771:
1767:
1764:
1755:
1745:
1738:
1733:
1727:
1723:
1720:
1713:
1710:
1703:
1700:
1691:
1685:
1681:
1674:
1667:
1650:
1645:
1641:
1628:
1626:
1624:
1580:
1529:th column of
1515:
1511:
1501:
1497:
1475:
1471:
1460:
1456:
1449:
1435:
1431:
1418:
1416:
1414:
1410:
1409:orthogonality
1406:
1402:
1397:
1394:
1390:
1386:
1382:
1377:
1375:
1367:
1362:
1360:
1358:
1354:
1350:
1346:
1342:
1334:
1332:
1310:
1302:
1296:
1291:
1286:
1279:
1274:
1269:
1262:
1259:
1254:
1249:
1243:
1234:
1230:
1221:
1217:
1213:
1208:
1204:
1200:
1197:
1193:
1187:
1181:
1176:
1171:
1164:
1159:
1154:
1147:
1142:
1137:
1131:
1120:
1116:
1107:
1103:
1099:
1094:
1090:
1085:
1082:
1072:
1066:
1063:
1058:
1055:
1050:
1043:
1038:
1033:
1026:
1021:
1016:
1010:
1001:
997:
988:
984:
980:
975:
971:
967:
964:
960:
954:
948:
943:
938:
931:
926:
921:
914:
909:
904:
898:
889:
885:
876:
872:
868:
863:
859:
855:
851:
848:
841:
835:
830:
825:
818:
813:
810:
805:
802:
795:
790:
785:
779:
751:
745:
740:
735:
728:
723:
720:
715:
712:
705:
700:
695:
689:
684:
681:
668:
666:
662:
658:
652:
644:
639:
637:
633:
629:
619:
593:
587:
584:
579:
572:
567:
560:
557:
552:
545:
542:
537:
531:
526:
515:
507:
489:
483:
480:
475:
470:
467:
462:
459:
452:
447:
442:
437:
431:
426:
423:
414:
412:
394:
388:
383:
378:
371:
366:
361:
354:
349:
344:
338:
324:
322:
308:
288:
256:
253:
247:
244:
241:
229:
226:
218:
194:
189:
187:
183:
179:
174:
168:
164:
159:
153:
145:
137:
133:
125:
120:
118:
110:
108:
105:
97:
89:
82:
76:
72:
68:
60:
56:
52:
48:
41:
37:
33:
19:
6576:Vector space
6439:
6308:Vector space
6204:
6170:
6143:
6117:
6095:
6065:
6035:
6003:
5997:
5979:
5973:
5949:
5943:
5937:
5925:
5920:p. 71, § 4.3
5913:
5901:
5889:
5873:
5867:
5850:
5845:
5838:Roman (2005)
5823:Axler (2015)
5818:
5438:
5334:
5327:Halmos (1974
5288:
5254:Matroid rank
5239:
5227:tensor order
5224:
5200:
5189:
5182:
5170:
5162:controllable
5151:
5123:
5120:Applications
4932:
4754:
4616:real numbers
4399:
4393:
4387:
4294:
4290:
4280:
4276:
4264:
4254:
4250:
4248:denotes the
4243:
4239:
4154:
4150:
4140:
4136:
4130:is equal to
4126:The rank of
4060:
4056:
3984:
3980:
3885:
3881:
3848:
3844:
3832:
3806:
3773:
3769:
3763:
3759:
3685:matrix is a
3681:
3677:
3667:
3664:
3660:
3656:
3646:
3642:
3635:
3566:The rank of
3565:
3541:
3525:
3507:
3503:
3496:
3482:The rank of
3481:
3422:The rank of
3421:
3406:
3402:
3396:
3392:
3374:
3370:
3366:
3363:
3322:
3137:
3011:there exist
2792:there exist
2769:
2765:
2755:
2751:
2704:The rank of
2703:
2680:The rank of
2679:
2656:
2648:column space
2569:The rank of
2568:
2540:
2493:The rank of
2382:
2364:
2360:
2353:
2314:is a set of
2309:
2305:
2302:
2295:
2292:
2285:
2282:
2271:
2267:
2264:
2261:
2254:
2250:
2247:
2240:
2237:
2230:
2227:
2217:
2210:
2203:
2192:
2188:
2081:
2074:
2068:
2050:
2043:
2040:
2030:
2020:
2013:
2009:
2003:
1996:
1990:
1983:
1977:
1973:
1750:
1743:
1736:
1725:
1721:
1718:
1711:
1708:
1701:
1698:
1683:
1679:
1672:
1665:
1649:real numbers
1643:
1639:
1632:
1621:. (Also see
1577:and, by the
1513:
1509:
1499:
1495:
1473:
1469:
1458:
1454:
1447:
1433:
1429:
1422:
1413:real numbers
1398:
1378:
1371:
1338:
669:
654:
631:
627:
415:
328:
195:or operator
190:
185:
181:
177:
175:
166:
160:
143:
142:, while the
136:column space
123:
121:
114:
103:
95:
87:
83:
51:vector space
35:
29:
6556:Multivector
6521:Determinant
6478:Dot product
6323:Linear span
5932:p. 90, § 50
5207:smooth maps
5196:tensor rank
4940:(i.e., the
4750:null spaces
4624:Gram matrix
4284:matrix and
3781:zero matrix
3700:. That is,
3687:nonnegative
3618:tensor rank
3562:Tensor rank
2759:matrix and
2452:defined by
1537:columns of
1489:columns in
1335:Computation
416:The matrix
329:The matrix
124:column rank
77:encoded by
61:columns of
18:Column rank
6590:Direct sum
6425:Invertible
6328:Linear map
5849:Bourbaki,
5810:References
5215:derivative
5166:observable
4758:for which
4162:such that
3868:(that is,
3858:invertible
3817:surjective
3671:as above.
3632:Properties
3574:such that
3400:and a map
3386:such that
2967:such that
2712:such that
2262:Now, each
2056:orthogonal
1549:(which is
1507:such that
1445:, and let
193:linear map
6620:Numerical
6383:Transpose
6064:(1974) .
5966:218542661
5775:
5769:−
5754:
5699:
5693:−
5681:
5606:
5600:≤
5585:
5529:
5506:
5500:→
5485:
5459:
5410:
5404:
5386:
5380:
5374:≤
5359:
5353:
5306:Φ
5300:ρ
5096:∗
5082:
5065:∗
5054:
5040:∗
5029:
5002:
4988:¯
4977:
4959:
4912:¯
4715:
4697:
4667:
4637:
4559:
4541:
4535:≤
4517:
4474:
4456:
4450:≤
4435:
4414:
4383:Frobenius
4351:
4345:≤
4339:−
4327:
4309:
4266:Sylvester
4100:
4079:
4024:
4003:
3945:
3927:
3915:≤
3900:
3864:has rank
3823:has rank
3797:has rank
3791:injective
3770:full rank
3723:≤
3711:
3593:⋅
3459:∗
3451:Σ
3340:⇔
3289:…
3239:…
3191:⇔
3179:⇔
3167:⇔
3155:⇔
3096:…
3038:…
2948:×
2918:×
2877:…
2819:…
2690:row space
2644:dimension
2611:…
2430:→
2143:⋯
1917:⋯
1827:⋯
1260:−
1227:→
1198:−
1113:→
1064:−
1056:−
994:→
965:−
882:→
811:−
803:−
721:−
713:−
674:given by
630:) = rank(
618:transpose
585:−
558:−
543:−
506:transpose
481:−
468:−
460:−
263:Φ
257:
248:
236:Φ
230:
203:Φ
178:full rank
152:row space
132:dimension
69:" of the
47:dimension
6681:Category
6666:Category
6605:Subspace
6600:Quotient
6551:Bivector
6465:Bilinear
6407:Matrices
6282:Glossary
6179:Springer
6152:Springer
6142:(2005).
6094:(2020).
6074:Springer
6044:Springer
6034:(2015).
5248:See also
5209:between
4948:), then
3852:), then
3501:-minor (
3369: :
3065:of size
2846:of size
2796:columns
2745:, where
1565:rows of
1194:→
1086:→
961:→
852:→
325:Examples
144:row rank
6277:Outline
6025:Sources
5851:Algebra
5192:tensors
4942:adjoint
4601:nullity
4298:, then
4158:matrix
4144:matrix
4068:, then
3992:, then
3879:is any
3779:Only a
3690:integer
3536:, then
3520:, then
2963:matrix
2933:matrix
2646:of the
2557:. The
1663:. Let
1503:matrix
1477:matrix
1453:, ...,
134:of the
130:is the
55:spanned
49:of the
45:is the
6561:Tensor
6373:Kernel
6303:Vector
6298:Scalar
6211:
6185:
6158:
6128:
6102:
6080:
6050:
6010:
5986:
5964:
5231:tensor
5194:, the
4274:is an
4237:where
4054:is an
3978:is an
3640:is an
3526:subset
2996:(when
2937:and a
2749:is an
2551:kernel
2280:. So,
2216:= ⋯ =
2002:+ ⋯ +
1971:where
1637:be an
1427:be an
1331:is 2.
1308:
665:pivots
281:where
40:matrix
34:, the
6430:Minor
6415:Block
6353:Basis
5962:S2CID
5321:from
5280:Notes
5185:rings
5164:, or
4580:when
4385:: if
4270:: if
3488:minor
3015:rows
2763:is a
2370:field
2301:, …,
2246:, …,
1749:, …,
1717:, …,
1690:basis
1688:be a
1678:, …,
1405:field
1383:. As
659:, by
626:rank(
217:image
102:rank
86:rank(
38:of a
6585:Dual
6440:Rank
6209:ISBN
6183:ISBN
6156:ISBN
6126:ISBN
6100:ISBN
6078:ISBN
6048:ISBN
6008:ISBN
5984:ISBN
5772:rank
5751:rank
5696:rank
5678:rank
5205:for
5203:rank
5079:rank
5051:rank
5026:rank
4999:rank
4974:rank
4956:rank
4930:and
4899:and
4712:rank
4694:rank
4664:rank
4634:rank
4584:and
4556:rank
4538:rank
4514:rank
4471:rank
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4432:rank
4411:rank
4397:and
4348:rank
4324:rank
4306:rank
4097:rank
4076:rank
4021:rank
4000:rank
3942:rank
3924:rank
3897:rank
3831:has
3805:has
3758:min(
3708:rank
3663:) =
3560:and
1730:are
1633:Let
1423:Let
227:rank
167:rank
122:The
73:and
36:rank
5954:doi
5878:doi
5603:dim
5582:dim
5526:ker
5503:ker
5482:ker
5456:ker
5407:ker
5401:dim
5383:ker
5377:dim
5356:ker
5350:dim
5160:is
5152:In
4944:of
4891:If
4610:If
4394:ABC
4288:is
4050:If
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3696:or
3654:by
3542:and
3490:in
3004:of
2692:of
2672:).
2650:of
2638:of
2553:of
2223:= 0
2077:= 0
2054:is
2046:= 0
1659:is
1625:.)
1441:be
620:of
608:of
309:img
289:dim
254:img
245:dim
169:of
154:of
146:of
138:of
126:of
94:rk(
92:or
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6683::
6181:.
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6146:.
6124:.
6120:.
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6068:.
6046:.
6038:.
5960:,
5950:78
5948:,
5874:68
5872:,
5858:^
5830:^
5217:.
5168:.
4789:0.
4607:.)
4400:BC
4391:,
4388:AB
4293:×
4279:×
4253:×
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3847:=
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3809:).
3762:,
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3645:×
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3506:×
3405:→
3395:→
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3008:),
2768:×
2754:×
2696:.
2375:.
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2236:,
2209:=
2174:0.
2085:,
1989:+
1976:=
1757::
1742:,
1707:,
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1517:.
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1512:=
1498:×
1472:×
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636:.
242::=
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158:.
107:.
6258:e
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6244:v
6217:.
6191:.
6164:.
6134:.
6108:.
6086:.
6056:.
6016:.
5956::
5880::
5802:.
5790:)
5787:C
5784:B
5781:A
5778:(
5766:)
5763:B
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5757:(
5731:A
5711:)
5708:C
5705:B
5702:(
5690:)
5687:B
5684:(
5658:B
5638:C
5635:B
5615:)
5612:M
5609:(
5597:)
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5591:A
5588:(
5562:M
5538:)
5535:B
5532:(
5522:/
5518:)
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5509:(
5497:)
5494:C
5491:B
5488:(
5478:/
5474:)
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5422:.
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5416:B
5413:(
5398:+
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5392:A
5389:(
5371:)
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5365:A
5362:(
5309:)
5303:(
5146:k
5142:k
5104:.
5101:)
5092:A
5088:A
5085:(
5076:=
5073:)
5070:A
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5057:(
5048:=
5045:)
5036:A
5032:(
5023:=
5020:)
5014:T
5009:A
5005:(
4996:=
4993:)
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4980:(
4971:=
4968:)
4965:A
4962:(
4946:A
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4933:A
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4909:A
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4877:.
4872:2
4867:|
4862:x
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4854:|
4849:=
4845:x
4841:A
4835:T
4830:A
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4812:=
4809:0
4786:=
4782:x
4778:A
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4767:A
4755:x
4736:.
4733:)
4727:T
4722:A
4718:(
4709:=
4706:)
4703:A
4700:(
4691:=
4688:)
4682:T
4677:A
4673:A
4670:(
4661:=
4658:)
4655:A
4649:T
4644:A
4640:(
4620:A
4612:A
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4590:k
4586:B
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4562:(
4553:+
4550:)
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4544:(
4532:)
4529:B
4526:+
4523:A
4520:(
4492:.
4489:)
4486:C
4483:B
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4477:(
4468:+
4465:)
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4459:(
4447:)
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4366:.
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4360:B
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4354:(
4342:n
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4330:(
4321:+
4318:)
4315:A
4312:(
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4286:B
4281:n
4277:m
4272:A
4261:.
4255:r
4251:r
4244:r
4240:I
4225:,
4220:]
4214:0
4209:0
4202:0
4195:r
4191:I
4184:[
4179:=
4176:Y
4173:A
4170:X
4160:Y
4155:n
4151:n
4146:X
4141:m
4137:m
4132:r
4128:A
4112:.
4109:)
4106:A
4103:(
4094:=
4091:)
4088:A
4085:C
4082:(
4066:m
4061:m
4057:l
4052:C
4036:.
4033:)
4030:A
4027:(
4018:=
4015:)
4012:B
4009:A
4006:(
3990:n
3985:k
3981:n
3976:B
3960:.
3957:)
3954:)
3951:B
3948:(
3939:,
3936:)
3933:A
3930:(
3921:(
3912:)
3909:B
3906:A
3903:(
3886:k
3882:n
3877:B
3870:A
3866:n
3862:A
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3803:A
3799:n
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3787:f
3776:.
3766:)
3764:n
3760:m
3744:.
3741:)
3738:n
3735:,
3732:m
3729:(
3720:)
3717:A
3714:(
3698:n
3694:m
3682:n
3678:m
3668:x
3665:A
3661:x
3659:(
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3643:m
3638:A
3614:r
3610:c
3596:r
3590:c
3580:k
3576:A
3572:k
3568:A
3546:p
3538:p
3534:p
3530:n
3522:p
3518:p
3514:n
3508:p
3504:p
3499:p
3492:A
3484:A
3473:.
3455:V
3448:U
3445:=
3442:A
3424:A
3407:W
3403:X
3397:X
3393:V
3388:f
3384:X
3380:k
3375:W
3371:V
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3349:)
3346:5
3343:(
3337:)
3334:1
3331:(
3318:C
3302:k
3297:c
3292:,
3286:,
3281:1
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3252:k
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3242:,
3236:,
3231:1
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3214:C
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3197:5
3194:(
3188:)
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3182:(
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3132:k
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3099:,
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3088:1
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3051:k
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3035:,
3030:1
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3013:k
3006:A
2998:k
2984:R
2981:C
2978:=
2975:A
2965:R
2951:n
2945:k
2935:C
2921:k
2915:m
2904:,
2890:k
2885:c
2880:,
2874:,
2869:1
2864:c
2852:A
2848:m
2832:k
2827:c
2822:,
2816:,
2811:1
2806:c
2794:k
2789:,
2787:k
2783:A
2775:k
2770:n
2766:k
2761:R
2756:k
2752:m
2747:C
2733:R
2730:C
2727:=
2724:A
2714:A
2710:k
2706:A
2694:A
2686:A
2682:A
2670:A
2666:f
2662:A
2657:F
2652:A
2640:A
2624:k
2619:c
2614:,
2608:,
2603:2
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2481:.
2478:x
2475:A
2472:=
2469:)
2466:x
2463:(
2460:f
2438:m
2434:F
2425:n
2421:F
2417::
2414:f
2391:A
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2356:A
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2328:A
2324:A
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2306:x
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2251:x
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2241:x
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2228:A
2220:r
2218:c
2214:2
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2207:1
2204:c
2199:A
2193:i
2189:x
2171:=
2166:r
2161:x
2154:r
2150:c
2146:+
2140:+
2135:2
2130:x
2123:2
2119:c
2115:+
2110:1
2105:x
2098:1
2094:c
2082:v
2075:v
2069:v
2064:A
2060:A
2051:v
2044:v
2041:A
2036:A
2031:v
2026:A
2021:v
2014:r
2010:x
2006:r
2004:c
2000:2
1997:x
1994:2
1991:c
1987:1
1984:x
1981:1
1978:c
1974:v
1959:,
1955:v
1951:A
1948:=
1945:)
1940:r
1935:x
1928:r
1924:c
1920:+
1914:+
1909:2
1904:x
1897:2
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1889:+
1884:1
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1872:1
1868:c
1864:(
1861:A
1858:=
1853:r
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1830:+
1824:+
1819:2
1814:x
1809:A
1804:2
1800:c
1796:+
1791:1
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1781:A
1776:1
1772:c
1768:=
1765:0
1753:r
1751:c
1747:2
1744:c
1740:1
1737:c
1726:r
1722:x
1719:A
1715:2
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1705:1
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1699:A
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1684:r
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1500:n
1496:r
1491:C
1487:r
1483:A
1479:C
1474:r
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1459:r
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1448:c
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1430:m
1425:A
1329:A
1311:.
1303:]
1297:0
1292:0
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1280:3
1275:1
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1222:1
1218:R
1214:+
1209:2
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1201:2
1188:]
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1132:[
1121:3
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1100:+
1095:2
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922:0
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910:2
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877:2
873:R
869:+
864:1
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842:]
836:0
831:5
826:3
819:1
814:3
806:2
796:1
791:2
786:1
780:[
752:]
746:0
741:5
736:3
729:1
724:3
716:2
706:1
701:2
696:1
690:[
685:=
682:A
672:A
634:)
632:A
628:A
622:A
614:A
610:A
594:]
588:2
580:2
573:0
568:0
561:1
553:1
546:1
538:1
532:[
527:=
521:T
516:A
490:]
484:2
476:0
471:1
463:1
453:2
448:0
443:1
438:1
432:[
427:=
424:A
395:]
389:1
384:1
379:0
372:1
367:1
362:0
355:1
350:0
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339:[
269:)
266:)
260:(
251:(
239:)
233:(
219::
171:A
156:A
148:A
140:A
128:A
104:A
98:)
96:A
90:)
88:A
79:A
63:A
43:A
20:)
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