6004:
2338:
39:
2074:
3023:
1423:
3745:
players plays one match against each of the other players, each match resulting in a win for one of the players and a loss for the other. A tournament is regular if each player wins the same number of matches. A regular tournament is doubly regular if the number of opponents beaten by both of two
3528:
if one can be obtained from the other by negating rows or columns, or by interchanging rows or columns. Up to equivalence, there is a unique
Hadamard matrix of orders 1, 2, 4, 8, and 12. There are 5 inequivalent matrices of order 16, 3 of order 20, 60 of order 24, and 487 of order 28. Millions of
2542:
2333:{\displaystyle {\begin{aligned}H_{1}&={\begin{bmatrix}1\end{bmatrix}},\\H_{2}&={\begin{bmatrix}1&1\\1&-1\end{bmatrix}},\\H_{4}&={\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{bmatrix}},\end{aligned}}}
2824:
88:
terms, it means that each pair of rows has matching entries in exactly half of their columns and mismatched entries in the remaining columns. It is a consequence of this definition that the corresponding properties hold for columns as well as rows.
3721:
A skew
Hadamard matrix remains a skew Hadamard matrix after multiplication of any row and its corresponding column by −1. This makes it possible, for example, to normalize a skew Hadamard matrix so that all elements in the first row equal 1.
1107:
3798:− 3)/4. A skew Hadamard matrix is obtained by introducing an additional player who defeats all of the original players and then forming a matrix with rows and columns labeled by players according to the rule that row
1858:
2349:
4167:
Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers.
2055:
2725:
3893:
The circulant
Hadamard matrix conjecture, however, asserts that, apart from the known 1 × 1 and 4 × 4 examples, no such matrices exist. This was verified for all but 26 values of
3512:
In 2005, Hadi
Kharaghani and Behruz Tayfeh-Rezaie published their construction of a Hadamard matrix of order 428. As a result, the smallest order for which no Hadamard matrix is presently known is 668.
3018:{\displaystyle {\begin{aligned}F_{1}&={\begin{bmatrix}0&1\end{bmatrix}}\\F_{n}&={\begin{bmatrix}0_{1\times 2^{n-1}}&1_{1\times 2^{n-1}}\\F_{n-1}&F_{n-1}\end{bmatrix}}.\end{aligned}}}
503:
385:
3830:. This correspondence in reverse produces a doubly regular tournament from a skew Hadamard matrix, assuming the skew Hadamard matrix is normalized so that all elements of the first row equal 1.
991:
777:
3100:
2829:
2079:
1672:
888:
3516:
By 2014, there were 12 multiples of 4 less than 2000 for which no
Hadamard matrix of that order was known. They are: 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, and 1964.
3450:
Sylvester's 1867 construction yields
Hadamard matrices of order 1, 2, 4, 8, 16, 32, etc. Hadamard matrices of orders 12 and 20 were subsequently constructed by Hadamard (in 1893). In 1933,
1793:
1897:
245:
1059:
3719:
3949:
1712:
3441:
1967:
1752:
3628:
1594:
1924:
1620:
314:
2785:
2577:
1418:{\displaystyle \sum _{i=0}^{n-1}a_{k,i}a_{l,i}=\sum _{i=0}^{n-1}h_{0,j}h_{k,i}h_{0,j}h_{l,i}=\sum _{i=0}^{n-1}h_{0,j}^{2}h_{k,i}h_{l,i}=\sum _{i=0}^{n-1}h_{k,i}h_{l,i}=0.}
5662:
3288:
3137:
677:
3489:
The smallest order that cannot be constructed by a combination of
Sylvester's and Paley's methods is 92. A Hadamard matrix of this order was found using a computer by
2597:
3211:
1565:
1545:
1505:
803:
651:
3244:
1525:
625:
567:
3393:
3366:
3164:
2816:
2752:
1482:
1454:
4290:
3648:
3578:
3558:
1099:
1079:
4617:
5069:
JPL: In 1961, mathematicians from NASA’s Jet
Propulsion Laboratory and Caltech worked together to construct a Hadamard Matrix containing 92 rows and columns
4769:
5876:
4020:– an amateur-radio digital protocol designed to work in difficult (low signal-to-noise ratio plus multipath propagation) conditions on shortwave bands.
6050:
131:
2628:
zero. The elements in the first column and the first row are all positive. The elements in all the other rows and columns are evenly divided between
5095:
3316:
5967:
5068:
1799:
5007:
4758:
5886:
5652:
2537:{\displaystyle H_{2^{k}}={\begin{bmatrix}H_{2^{k-1}}&H_{2^{k-1}}\\H_{2^{k-1}}&-H_{2^{k-1}}\end{bmatrix}}=H_{2}\otimes H_{2^{k-1}},}
4257:
2004:
3794:− 1 other players together defeat the same number of common opponents. This common number of defeated opponents must therefore be (
3338:. The Hadamard conjecture has also been attributed to Paley, although it was considered implicitly by others prior to Paley's work.
2650:
81:
5687:
3506:
5234:
4559:
4399:Đoković, Dragomir Ž; Golubitsky, Oleg; Kotsireas, Ilias S. (2014). "Some new orders of Hadamard and Skew-Hadamard matrices".
442:
330:
6040:
4060:
4023:
3990:
3861:
matrix is manifestly regular, and therefore a circulant
Hadamard matrix would have to be of square order. Moreover, if an
3498:
896:
682:
150:
4236:
3038:
5451:
5088:
4749:
Georgiou, S.; Koukouvinos, C.; Seberry, J. (2003). "Hadamard matrices, orthogonal designs and construction algorithms".
3841:
are real
Hadamard matrices whose row and column sums are all equal. A necessary condition on the existence of a regular
1625:
811:
5526:
4050:
1760:
5682:
5204:
1864:
3790:− 3)/4. The same result should be obtained if the pairs are counted differently: the given player and any of the
5786:
5657:
5571:
4329:
3838:
3726:
3509:, that has yielded many additional orders. Many other methods for constructing Hadamard matrices are now known.
5891:
5781:
5489:
5169:
5063:
4803:
433:
202:
5926:
5855:
5737:
5597:
5194:
5081:
3955:
996:
324:
whose transpose is thus its inverse. Multiplying by the length again gives the equality above. As a result,
3677:
3630:
entries randomly deleted, then with overwhelming likelihood, one can perfectly recover the original matrix
3447:. This result is used to produce Hadamard matrices of higher order once those of smaller orders are known.
5796:
5379:
5184:
4169:
4047:
Feedback delay networks – Digital reverberation devices which use Hadamard matrices to blend sample values
3029:
2625:
2064:. This observation can be applied repeatedly and leads to the following sequence of matrices, also called
1987:
1970:
3913:
1677:
5742:
5479:
5329:
5324:
5159:
5134:
5129:
4031:
3406:
3167:
2727:, we can describe an alternative construction of Sylvester's Hadamard matrix. First consider the matrix
1932:
1717:
569:, then there is at least one scalar product of 2 rows which has to be 0. The scalar product is a sum of
138:
6003:
146:
1570:
Row 2 has to be orthogonal to row 1, so the number of products of entries of the rows resulting in 1,
540:
of the nonexistence of Hadamard matrices with dimensions other than 1, 2, or a multiple of 4 follows:
5936:
5294:
5124:
5104:
4188:
4097:
4054:
3650:
from the damaged one. The algorithm of recovery has the same computational cost as matrix inversion.
3583:
3537:, there are 4 inequivalent matrices of order 16, 3 of order 20, 36 of order 24, and 294 of order 28.
3530:
3525:
3490:
1573:
586:
574:
127:
1903:
1599:
291:
5957:
5931:
5509:
5314:
5304:
3986:
3974:
3951:
for some w, its weight. A weighing matrix with its weight equal to its order is a Hadamard matrix.
2757:
281:
2550:
6008:
5962:
5952:
5906:
5901:
5830:
5766:
5632:
5369:
5364:
5299:
5289:
5154:
4959:
4820:
4789:
4693:
4646:
4502:
4461:
4426:
4408:
4381:
4207:
4122:
4102:
4084:
4066:
3471:
3455:
2645:
2629:
537:
4141:
4044:. The mask element used in coded aperture spectrometers is often a variant of a Hadamard matrix.
3540:
Hadamard matrices are also uniquely recoverable, in the following sense: If an Hadamard matrix
3658:
Many special cases of Hadamard matrices have been investigated in the mathematical literature.
3259:
3108:
656:
6045:
6019:
5806:
5801:
5791:
5771:
5732:
5727:
5556:
5551:
5536:
5531:
5522:
5517:
5464:
5359:
5309:
5254:
5224:
5219:
5199:
5189:
5149:
5003:
4754:
4685:
4080:
4070:
4041:
3494:
2600:
2582:
321:
3910:. A weighing matrix is a square matrix in which entries may also be zero and which satisfies
3183:
1550:
1530:
1490:
782:
630:
6014:
5982:
5911:
5850:
5845:
5825:
5761:
5667:
5637:
5622:
5607:
5602:
5541:
5494:
5469:
5459:
5430:
5349:
5344:
5319:
5249:
5229:
5139:
5119:
4984:
4951:
4926:
4899:
4872:
4845:
4812:
4781:
4727:
4677:
4636:
4626:
4568:
4533:
4492:
4453:
4418:
4373:
4338:
4299:
4266:
4197:
3982:
3219:
2617:
1510:
595:
546:
62:
4741:
4582:
4350:
4313:
4219:
3371:
3344:
3142:
2794:
2730:
5712:
5647:
5627:
5612:
5592:
5576:
5474:
5405:
5395:
5354:
5209:
5042:
4737:
4578:
4346:
4309:
4215:
4117:
4107:
3907:
2606:
In this manner, Sylvester constructed Hadamard matrices of order 2 for every non-negative
265:
4053:
of experiments for investigating the dependence of some measured quantity on a number of
1459:
1431:
130:
with entries of absolute value less than or equal to 1 and so is an extremal solution of
5972:
5916:
5896:
5881:
5840:
5717:
5677:
5642:
5566:
5505:
5484:
5425:
5415:
5400:
5334:
5279:
5269:
5264:
5174:
4037:
3997:
3959:
3633:
3563:
3543:
2633:
1084:
1064:
512:
406:
193:
119:
97:
42:
5053:
4732:
4715:
38:
6034:
5977:
5835:
5776:
5707:
5697:
5692:
5617:
5546:
5420:
5410:
5339:
5259:
5244:
5179:
4989:
4972:
4930:
4904:
4887:
4877:
4860:
4697:
4538:
4521:
4506:
4465:
4327:
Williamson, J. (1944). "Hadamard's determinant theorem and the sum of four squares".
3854:
3451:
3253:
142:
85:
78:
66:
4824:
4793:
4430:
4385:
4304:
4285:
5860:
5817:
5722:
5435:
5374:
5284:
5164:
4112:
4076:
3746:
distinct players is the same for all pairs of distinct players. Since each of the
3483:
3323:
2065:
154:
31:
17:
4631:
4612:
4342:
4963:
3754:− 1)/2 matches played results in a win for one of the players, each player wins (
5702:
5672:
5440:
5274:
5144:
4017:
3467:
3214:
3178:
3171:
285:
123:
54:
5059:
4596:
Turyn, R. J. (1969). "Sequences with small correlation". In Mann, H. B. (ed.).
3032:
that the image of the Hadamard matrix under the above homomorphism is given by
5753:
5214:
4955:
4850:
4833:
4785:
4681:
4497:
4480:
4457:
3249:
70:
4689:
4202:
4183:
77:
terms, this means that each pair of rows in a Hadamard matrix represents two
5987:
5561:
4573:
4554:
4364:
Kharaghani, H.; Tayfeh-Rezaie, B. (2005). "A Hadamard matrix of order 428".
4234:
Hadamard, J. (1893). "Résolution d'une question relative aux déterminants".
3858:
3534:
185:
165:
162:
5030:
4270:
1853:{\displaystyle \alpha +{\frac {n}{2}}-\beta =\beta +{\frac {n}{2}}-\alpha }
3326:
in the theory of Hadamard matrices is one of existence. Specifically, the
5921:
4027:
158:
74:
4914:
4939:
4816:
4665:
4650:
4641:
4211:
3297:
2607:
5023:
4664:
Geramita, Anthony V.; Pullman, Norman J.; Wallis, Jennifer S. (1974).
4422:
4377:
4861:"Doubly regular tournaments are equivalent to skew Hadamard matrices"
4522:"Doubly regular tournaments are equivalent to skew hadamard matrices"
118:
among parallelotopes spanned by vectors whose entries are bounded in
115:
3105:
This construction demonstrates that the rows of the Hadamard matrix
3993:
are complex Hadamard matrices in which the entries are taken to be
3529:
inequivalent matrices are known for orders 32, 36, and 40. Using a
2616:
Sylvester's matrices have a number of special properties. They are
5026:
of all orders up to 100, including every type with order up to 28;
4751:
Designs 2002: Further computational and constructive design theory
4413:
3482:
is a prime power that is congruent to 1 modulo 4. His method uses
37:
5073:
5046:
3725:
Reid and Brown in 1972 showed that there exists a doubly regular
2791:-bit numbers arranged in ascending counting order. We may define
1986:
Examples of Hadamard matrices were actually first constructed by
521:
The order of a Hadamard matrix must be 1, 2, or a multiple of 4.
5056:
to obtain all orders up to 1000, except 668, 716, 876 & 892.
5036:
4004:
has been used by some authors to refer specifically to the case
3958:
to be a matrix in which the entries are complex numbers of unit
3758:− 1)/2 matches (and loses the same number). Since each of the (
5077:
4444:
Wanless, I.M. (2005). "Permanents of matrices of signed ones".
2050:{\displaystyle {\begin{bmatrix}H&H\\H&-H\end{bmatrix}}}
3502:
69:
whose entries are either +1 or −1 and whose rows are mutually
46:
3474:
to 3 modulo 4 and that produces a Hadamard matrix of order 2(
2720:{\displaystyle (\{1,-1\},\times )\mapsto (\{0,1\}),\oplus \}}
1428:
Row 1 and row 2, like all other rows except row 0, must have
573:
values each of which is either 1 or −1, therefore the sum is
3733:
if and only if there exists a skew Hadamard matrix of order
3341:
A generalization of Sylvester's construction proves that if
137:
Certain Hadamard matrices can almost directly be used as an
4801:
Kimura, Hiroshi (1989). "New Hadamard matrix of order 24".
1547:
denote the number of 1s of row 2 beneath −1s in row 1. Let
1527:
denote the number of −1s of row 2 beneath 1s in row 1. Let
4026:(BRR) – a technique used by statisticians to estimate the
1507:
denote the number of 1s of row 2 beneath 1s in row 1. Let
3762:− 1)/2 players defeated by a given player also loses to (
4973:"Classification of hadamard matrices of order 24 and 28"
2644:
If we map the elements of the Hadamard matrix using the
1929:
But we have as the number of 1s in row 1 the odd number
1567:
denote the number of −1s of row 2 beneath −1s in row 1.
3737: + 1. In a mathematical tournament of order
3256:, by contrast, is constructed from the Hadamard matrix
4073:
and under-determined linear systems (inverse problems)
2899:
2854:
2378:
2210:
2147:
2104:
2013:
498:{\displaystyle |\operatorname {det} (M)|\leq n^{n/2}.}
380:{\displaystyle \operatorname {det} (H)=\pm \,n^{n/2},}
4255:
Paley, R. E. A. C. (1933). "On orthogonal matrices".
3916:
3680:
3636:
3586:
3566:
3546:
3409:
3374:
3347:
3262:
3222:
3186:
3145:
3111:
3041:
2827:
2797:
2760:
2733:
2653:
2585:
2553:
2352:
2077:
2007:
1935:
1906:
1867:
1802:
1763:
1720:
1680:
1628:
1602:
1576:
1553:
1533:
1513:
1493:
1462:
1434:
1110:
1087:
1067:
999:
899:
814:
785:
685:
659:
633:
598:
549:
508:
Equality in this bound is attained for a real matrix
445:
333:
294:
205:
986:{\displaystyle A=(a_{i,j})_{i,j\in \{0,1,...,n-1\}}}
772:{\displaystyle H=(h_{i,j})_{i,j\in \{0,1,...,n-1\}}}
276:. To see that this is true, notice that the rows of
5945:
5869:
5815:
5751:
5585:
5503:
5449:
5388:
5112:
4063:
for investigating noise factor impacts on responses
3095:{\displaystyle H_{2^{n}}=F_{n}^{\textsf {T}}F_{n}.}
4915:"A construction for generalized hadamard matrices"
3981:. Complex Hadamard matrices arise in the study of
3943:
3766:− 3)/2 other players, the number of player pairs (
3713:
3642:
3622:
3572:
3552:
3435:
3387:
3360:
3282:
3238:
3205:
3158:
3131:
3094:
3017:
2810:
2779:
2746:
2719:
2591:
2571:
2536:
2332:
2049:
1961:
1918:
1891:
1852:
1787:
1746:
1706:
1667:{\displaystyle n/2=\alpha +\gamma =\beta +\delta }
1666:
1614:
1588:
1559:
1539:
1519:
1499:
1476:
1448:
1417:
1093:
1073:
1053:
985:
882:
797:
771:
671:
645:
619:
561:
497:
379:
308:
239:
122:by 1. Equivalently, a Hadamard matrix has maximal
4284:Baumert, L.; Golomb, S. W.; Hall, M. Jr. (1962).
883:{\displaystyle \sum _{i=0}^{n-1}h_{k,i}h_{l,i}=0}
4676:(1). Cambridge University Press (CUP): 119–122.
2632:. Sylvester matrices are closely connected with
4670:Bulletin of the Australian Mathematical Society
4040:spectrometry – an instrument for measuring the
4938:Seberry, J.; Wysocki, B.; Wysocki, T. (2005).
1788:{\displaystyle \alpha +\delta =\beta +\gamma }
5089:
4291:Bulletin of the American Mathematical Society
4286:"Discovery of an Hadamard Matrix of Order 92"
1892:{\displaystyle \alpha -\beta =\beta -\alpha }
8:
4618:Journal of the American Mathematical Society
3458:, which produces a Hadamard matrix of order
2714:
2702:
2690:
2672:
2657:
978:
942:
764:
728:
4940:"On some applications of Hadamard matrices"
2624: ≥ 1 (2 > 1), have
5663:Fundamental (linear differential equation)
5096:
5082:
5074:
4998:Yarlagadda, R. K.; Hershey, J. E. (1997).
4716:"Hadamard matrices of the Williamson type"
4184:"Hadamard matrices and their applications"
3877:would necessarily have to be of the form 4
3330:proposes that a Hadamard matrix of order 4
524:
4988:
4903:
4876:
4849:
4731:
4640:
4630:
4613:"Cyclotomic integers and finite geometry"
4572:
4537:
4526:Journal of Combinatorial Theory, Series A
4496:
4412:
4303:
4201:
3926:
3925:
3924:
3915:
3687:
3686:
3685:
3679:
3635:
3603:
3597:
3585:
3565:
3545:
3427:
3414:
3408:
3379:
3373:
3352:
3346:
3272:
3267:
3261:
3227:
3221:
3191:
3185:
3150:
3144:
3121:
3116:
3110:
3083:
3073:
3072:
3071:
3066:
3051:
3046:
3040:
2988:
2970:
2948:
2937:
2917:
2906:
2894:
2881:
2849:
2836:
2828:
2826:
2802:
2796:
2771:
2759:
2738:
2732:
2652:
2584:
2552:
2517:
2512:
2499:
2470:
2465:
2442:
2437:
2415:
2410:
2390:
2385:
2373:
2362:
2357:
2351:
2205:
2192:
2142:
2129:
2099:
2086:
2078:
2076:
2008:
2006:
1939:
1934:
1905:
1866:
1834:
1809:
1801:
1762:
1730:
1719:
1690:
1679:
1632:
1627:
1601:
1575:
1552:
1532:
1512:
1492:
1466:
1461:
1438:
1433:
1397:
1381:
1365:
1354:
1335:
1319:
1309:
1298:
1282:
1271:
1252:
1236:
1220:
1204:
1188:
1177:
1158:
1142:
1126:
1115:
1109:
1086:
1066:
1039:
1023:
1004:
998:
929:
913:
898:
862:
846:
830:
819:
813:
784:
715:
699:
684:
658:
632:
597:
548:
482:
478:
466:
446:
444:
364:
360:
355:
332:
302:
295:
293:
231:
215:
214:
213:
204:
108:Hadamard matrix has the maximum possible
4481:"Geometric search for Hadamard matrices"
779:, then it has the property that for any
49:in 2005, using Sylvester's construction.
45:demonstrates the Hadamard conjecture at
5968:Matrix representation of conic sections
4888:"On the existence of Hadamard matrices"
4768:Goethals, J. M.; Seidel, J. J. (1970).
4714:Baumert, L. D.; Hall, Marshall (1965).
4133:
3869:circulant Hadamard matrix existed with
3533:notion of equivalence that also allows
3317:(more unsolved problems in mathematics)
240:{\displaystyle HH^{\textsf {T}}=nI_{n}}
61:, named after the French mathematician
5000:Hadamard Matrix Analysis and Synthesis
1596:, has to match those resulting in −1,
1054:{\displaystyle a_{i,j}=h_{0,j}h_{i,j}}
132:Hadamard's maximal determinant problem
3714:{\displaystyle H^{\textsf {T}}+H=2I.}
3524:Two Hadamard matrices are considered
3305:Is there a Hadamard matrix of order 4
7:
5062:to generate Hadamard Matrices using
4770:"A skew Hadamard matrix of order 36"
4753:. Boston: Kluwer. pp. 133–205.
4600:. New York: Wiley. pp. 195–228.
4555:"Character sums and difference sets"
2787:matrix whose columns consist of all
280:are all orthogonal vectors over the
4237:Bulletin des Sciences Mathématiques
4182:Hedayat, A.; Wallis, W. D. (1978).
3944:{\displaystyle WW^{\textsf {T}}=wI}
3505:. They used a construction, due to
3290:by a slightly different procedure.
3248:This code is also referred to as a
1707:{\displaystyle \gamma =n/2-\alpha }
1081:has all 1s in row 0. We check that
4258:Journal of Mathematics and Physics
3436:{\displaystyle H_{n}\otimes H_{m}}
3334:exists for every positive integer
1962:{\displaystyle n/2=\alpha +\beta }
1747:{\displaystyle \delta =n/2-\beta }
25:
4886:Seberry Wallis, Jennifer (1976).
4838:Annals of Mathematical Statistics
4834:"On Hotelling's Weighing Problem"
4733:10.1090/S0025-5718-1965-0179093-2
3954:Another generalization defines a
6051:Unsolved problems in mathematics
6002:
4520:Reid, K.B.; Brown, Ezra (1972).
4401:Journal of Combinatorial Designs
4366:Journal of Combinatorial Designs
3395:are Hadamard matrices of orders
413:, whose entries are bounded by |
320:through by this length gives an
5870:Used in science and engineering
4859:Reid, K. B.; Brown, E. (1972).
4666:"Families of weighing matrices"
4305:10.1090/S0002-9904-1962-10761-7
4142:"Hadamard Matrices and Designs"
3623:{\displaystyle O(n^{2}/\log n)}
3300:Unsolved problem in mathematics
2060:is a Hadamard matrix of order 2
1589:{\displaystyle \alpha +\delta }
5113:Explicitly constrained entries
5047:"Library of Hadamard Matrices"
4560:Pacific Journal of Mathematics
4446:Linear and Multilinear Algebra
3906:One basic generalization is a
3617:
3590:
3443:is a Hadamard matrix of order
2705:
2687:
2684:
2681:
2654:
1998:. Then the partitioned matrix
1994:be a Hadamard matrix of order
1919:{\displaystyle \alpha =\beta }
1615:{\displaystyle \beta +\gamma }
926:
906:
712:
692:
467:
463:
457:
447:
346:
340:
309:{\displaystyle {\sqrt {n}}\,.}
180:be a Hadamard matrix of order
1:
5887:Fundamental (computer vision)
4632:10.1090/S0894-0347-99-00298-2
4343:10.1215/S0012-7094-44-01108-7
4024:Balanced repeated replication
3991:Butson-type Hadamard matrices
2780:{\displaystyle n\times 2^{n}}
151:balanced repeated replication
4990:10.1016/0012-365X(93)E0169-5
4931:10.1016/0378-3758(80)90021-X
4905:10.1016/0097-3165(76)90062-5
4878:10.1016/0097-3165(72)90098-2
4539:10.1016/0097-3165(72)90098-2
4485:Theoretical Computer Science
3782:and to the given player is (
2572:{\displaystyle 2\leq k\in N}
1981:
1674:, from which we can express
434:Hadamard's determinant bound
5653:Duplication and elimination
5452:eigenvalues or eigenvectors
4832:Mood, Alexander M. (1964).
3889:Circulant Hadamard matrices
3309:for every positive integer
1622:. Due to (*), we also have
1101:is also a Hadamard matrix:
6067:
5586:With specific applications
5215:Discrete Fourier Transform
4913:Seberry, Jennifer (1980).
3520:Equivalence and uniqueness
3139:can be viewed as a length
192:is closely related to its
100:spanned by the rows of an
29:
5996:
5877:Cabibbo–Kobayashi–Maskawa
5504:Satisfying conditions on
4956:10.1007/s00184-005-0415-y
4786:10.1017/S144678870000673X
4682:10.1017/s0004972700040703
4498:10.1016/j.tcs.2019.01.025
4458:10.1080/03081080500093990
4330:Duke Mathematical Journal
3839:Regular Hadamard matrices
3834:Regular Hadamard matrices
3283:{\displaystyle H_{2^{n}}}
3132:{\displaystyle H_{2^{n}}}
893:Now we define the matrix
672:{\displaystyle n\times n}
4919:J. Statist. Plann. Infer
4865:J. Combin. Theory Ser. A
4804:Graphs and Combinatorics
4061:Robust parameter designs
3849:Hadamard matrix is that
2640:Alternative construction
2592:{\displaystyle \otimes }
1982:Sylvester's construction
1484:entries of −1 each. (*)
394:) is the determinant of
149:), and are also used in
5235:Generalized permutation
4971:Spence, Edward (1995).
4851:10.1214/aoms/1177730883
4574:10.2140/pjm.1965.15.319
4087:for quantum algorithms.
4002:complex Hadamard matrix
3956:complex Hadamard matrix
3206:{\displaystyle 2^{n-1}}
1560:{\displaystyle \delta }
1540:{\displaystyle \gamma }
1500:{\displaystyle \alpha }
798:{\displaystyle k\neq l}
646:{\displaystyle m\geq 1}
420: | ≤ 1, for each
6009:Mathematics portal
5024:Skew Hadamard matrices
4598:Error Correcting Codes
4271:10.1002/sapm1933121311
4203:10.1214/aos/1176344370
4170:Philosophical Magazine
4051:Plackett–Burman design
4012:Practical applications
3945:
3715:
3662:Skew Hadamard matrices
3644:
3624:
3574:
3554:
3437:
3389:
3362:
3284:
3240:
3239:{\displaystyle F_{n}.}
3207:
3160:
3133:
3096:
3019:
2812:
2781:
2748:
2721:
2593:
2573:
2538:
2334:
2051:
1988:James Joseph Sylvester
1963:
1920:
1893:
1854:
1789:
1748:
1708:
1668:
1616:
1590:
1561:
1541:
1521:
1520:{\displaystyle \beta }
1501:
1478:
1450:
1419:
1376:
1293:
1199:
1137:
1095:
1075:
1055:
987:
884:
841:
799:
773:
673:
653:, and there exists an
647:
621:
620:{\displaystyle n=4m+2}
563:
562:{\displaystyle n>1}
518:is a Hadamard matrix.
499:
381:
310:
241:
50:
4774:J. Austral. Math. Soc
4553:Turyn, R. J. (1965).
4077:Quantum Hadamard gate
4055:independent variables
4032:statistical estimator
3946:
3716:
3645:
3625:
3575:
3555:
3438:
3390:
3388:{\displaystyle H_{m}}
3363:
3361:{\displaystyle H_{n}}
3285:
3241:
3208:
3168:error-correcting code
3161:
3159:{\displaystyle 2^{n}}
3134:
3097:
3020:
2813:
2811:{\displaystyle F_{n}}
2782:
2749:
2747:{\displaystyle F_{n}}
2722:
2630:positive and negative
2594:
2574:
2539:
2335:
2052:
1964:
1921:
1894:
1855:
1790:
1749:
1709:
1669:
1617:
1591:
1562:
1542:
1522:
1502:
1479:
1451:
1420:
1350:
1267:
1173:
1111:
1096:
1076:
1056:
988:
885:
815:
800:
774:
674:
648:
622:
564:
500:
382:
311:
288:and each have length
242:
139:error-correcting code
41:
6041:Combinatorial design
4611:Schmidt, B. (1999).
4189:Annals of Statistics
4098:Combinatorial design
3962:and which satisfies
3914:
3678:
3634:
3584:
3564:
3544:
3407:
3372:
3345:
3260:
3220:
3184:
3143:
3109:
3039:
2825:
2795:
2758:
2731:
2651:
2583:
2551:
2350:
2075:
2005:
1933:
1904:
1865:
1800:
1761:
1718:
1678:
1626:
1600:
1574:
1551:
1531:
1511:
1491:
1460:
1432:
1108:
1085:
1065:
997:
897:
812:
783:
683:
657:
631:
596:
547:
443:
331:
292:
272:is the transpose of
203:
5958:Linear independence
5205:Diagonally dominant
3987:quantum computation
3975:conjugate transpose
3774: ) such that
3403:respectively, then
3328:Hadamard conjecture
3322:The most important
3294:Hadamard conjecture
3078:
3028:It can be shown by
1477:{\displaystyle n/2}
1449:{\displaystyle n/2}
1314:
27:Mathematics concept
18:Hadamard conjecture
5963:Matrix exponential
5953:Jordan normal form
5787:Fisher information
5658:Euclidean distance
5572:Totally unimodular
5002:. Boston: Kluwer.
4817:10.1007/BF01788676
4479:Kline, J. (2019).
4172:, 34:461–475, 1867
4123:Quantum logic gate
4103:Hadamard transform
4085:Hadamard transform
4067:Compressed sensing
3985:and the theory of
3941:
3711:
3666:A Hadamard matrix
3640:
3620:
3570:
3550:
3456:Paley construction
3433:
3385:
3358:
3280:
3236:
3203:
3156:
3129:
3092:
3062:
3015:
3013:
3002:
2867:
2808:
2777:
2744:
2717:
2646:group homomorphism
2589:
2569:
2534:
2486:
2330:
2328:
2317:
2175:
2112:
2047:
2041:
1959:
1916:
1889:
1850:
1785:
1744:
1704:
1664:
1612:
1586:
1557:
1537:
1517:
1497:
1474:
1446:
1415:
1294:
1091:
1071:
1051:
983:
880:
795:
769:
669:
643:
617:
559:
495:
377:
306:
237:
51:
6028:
6027:
6020:Category:Matrices
5892:Fuzzy associative
5782:Doubly stochastic
5490:Positive-definite
5170:Block tridiagonal
5031:"Hadamard Matrix"
5009:978-0-7923-9826-4
4892:J. Comb. Theory A
4760:978-1-4020-7599-5
4423:10.1002/jcd.21358
4378:10.1002/jcd.20043
4081:quantum computing
4071:signal processing
4042:spectrum of light
3983:operator algebras
3928:
3689:
3643:{\displaystyle H}
3573:{\displaystyle n}
3553:{\displaystyle H}
3215:generating matrix
3075:
2601:Kronecker product
1978:
1977:
1842:
1817:
1456:entries of 1 and
1094:{\displaystyle A}
1074:{\displaystyle A}
322:orthogonal matrix
300:
217:
147:Reed–Muller codes
16:(Redirected from
6058:
6015:List of matrices
6007:
6006:
5983:Row echelon form
5927:State transition
5856:Seidel adjacency
5738:Totally positive
5598:Alternating sign
5195:Complex Hadamard
5098:
5091:
5084:
5075:
5050:
5034:
5013:
4994:
4992:
4983:(1–3): 185–242.
4967:
4950:(2–3): 221–239.
4934:
4909:
4907:
4882:
4880:
4855:
4853:
4828:
4797:
4764:
4745:
4735:
4702:
4701:
4661:
4655:
4654:
4644:
4634:
4608:
4602:
4601:
4593:
4587:
4586:
4576:
4550:
4544:
4543:
4541:
4517:
4511:
4510:
4500:
4476:
4470:
4469:
4441:
4435:
4434:
4416:
4396:
4390:
4389:
4361:
4355:
4354:
4324:
4318:
4317:
4307:
4281:
4275:
4274:
4265:(1–4): 311–320.
4252:
4246:
4245:
4231:
4225:
4223:
4205:
4196:(6): 1184–1238.
4179:
4173:
4165:J.J. Sylvester.
4163:
4157:
4156:
4154:
4152:
4146:
4138:
3950:
3948:
3947:
3942:
3931:
3930:
3929:
3720:
3718:
3717:
3712:
3692:
3691:
3690:
3649:
3647:
3646:
3641:
3629:
3627:
3626:
3621:
3607:
3602:
3601:
3579:
3577:
3576:
3571:
3559:
3557:
3556:
3551:
3442:
3440:
3439:
3434:
3432:
3431:
3419:
3418:
3394:
3392:
3391:
3386:
3384:
3383:
3367:
3365:
3364:
3359:
3357:
3356:
3301:
3289:
3287:
3286:
3281:
3279:
3278:
3277:
3276:
3245:
3243:
3242:
3237:
3232:
3231:
3212:
3210:
3209:
3204:
3202:
3201:
3179:minimum distance
3165:
3163:
3162:
3157:
3155:
3154:
3138:
3136:
3135:
3130:
3128:
3127:
3126:
3125:
3101:
3099:
3098:
3093:
3088:
3087:
3077:
3076:
3070:
3058:
3057:
3056:
3055:
3024:
3022:
3021:
3016:
3014:
3007:
3006:
2999:
2998:
2981:
2980:
2961:
2960:
2959:
2958:
2930:
2929:
2928:
2927:
2886:
2885:
2872:
2871:
2841:
2840:
2817:
2815:
2814:
2809:
2807:
2806:
2786:
2784:
2783:
2778:
2776:
2775:
2753:
2751:
2750:
2745:
2743:
2742:
2726:
2724:
2723:
2718:
2598:
2596:
2595:
2590:
2578:
2576:
2575:
2570:
2543:
2541:
2540:
2535:
2530:
2529:
2528:
2527:
2504:
2503:
2491:
2490:
2483:
2482:
2481:
2480:
2455:
2454:
2453:
2452:
2428:
2427:
2426:
2425:
2403:
2402:
2401:
2400:
2369:
2368:
2367:
2366:
2339:
2337:
2336:
2331:
2329:
2322:
2321:
2197:
2196:
2180:
2179:
2134:
2133:
2117:
2116:
2091:
2090:
2056:
2054:
2053:
2048:
2046:
2045:
1968:
1966:
1965:
1960:
1943:
1925:
1923:
1922:
1917:
1898:
1896:
1895:
1890:
1859:
1857:
1856:
1851:
1843:
1835:
1818:
1810:
1794:
1792:
1791:
1786:
1754:and substitute:
1753:
1751:
1750:
1745:
1734:
1713:
1711:
1710:
1705:
1694:
1673:
1671:
1670:
1665:
1636:
1621:
1619:
1618:
1613:
1595:
1593:
1592:
1587:
1566:
1564:
1563:
1558:
1546:
1544:
1543:
1538:
1526:
1524:
1523:
1518:
1506:
1504:
1503:
1498:
1483:
1481:
1480:
1475:
1470:
1455:
1453:
1452:
1447:
1442:
1424:
1422:
1421:
1416:
1408:
1407:
1392:
1391:
1375:
1364:
1346:
1345:
1330:
1329:
1313:
1308:
1292:
1281:
1263:
1262:
1247:
1246:
1231:
1230:
1215:
1214:
1198:
1187:
1169:
1168:
1153:
1152:
1136:
1125:
1100:
1098:
1097:
1092:
1080:
1078:
1077:
1072:
1060:
1058:
1057:
1052:
1050:
1049:
1034:
1033:
1015:
1014:
992:
990:
989:
984:
982:
981:
924:
923:
889:
887:
886:
881:
873:
872:
857:
856:
840:
829:
804:
802:
801:
796:
778:
776:
775:
770:
768:
767:
710:
709:
679:Hadamard matrix
678:
676:
675:
670:
652:
650:
649:
644:
626:
624:
623:
618:
568:
566:
565:
560:
525:
504:
502:
501:
496:
491:
490:
486:
470:
450:
409:matrix of order
386:
384:
383:
378:
373:
372:
368:
315:
313:
312:
307:
301:
296:
246:
244:
243:
238:
236:
235:
220:
219:
218:
157:to estimate the
145:(generalized in
114:
63:Jacques Hadamard
21:
6066:
6065:
6061:
6060:
6059:
6057:
6056:
6055:
6031:
6030:
6029:
6024:
6001:
5992:
5941:
5865:
5811:
5747:
5581:
5499:
5445:
5384:
5185:Centrosymmetric
5108:
5102:
5054:On-line utility
5043:N. J. A. Sloane
5041:
5029:
5020:
5010:
4997:
4970:
4937:
4912:
4885:
4858:
4831:
4800:
4767:
4761:
4748:
4726:(91): 442–447.
4713:
4710:
4708:Further reading
4705:
4663:
4662:
4658:
4610:
4609:
4605:
4595:
4594:
4590:
4552:
4551:
4547:
4519:
4518:
4514:
4478:
4477:
4473:
4443:
4442:
4438:
4398:
4397:
4393:
4363:
4362:
4358:
4326:
4325:
4321:
4283:
4282:
4278:
4254:
4253:
4249:
4233:
4232:
4228:
4181:
4180:
4176:
4164:
4160:
4150:
4148:
4144:
4140:
4139:
4135:
4131:
4118:Weighing matrix
4108:Quincunx matrix
4094:
4014:
3967:
3920:
3912:
3911:
3908:weighing matrix
3904:
3902:Generalizations
3891:
3865: ×
3845: ×
3836:
3681:
3676:
3675:
3664:
3656:
3632:
3631:
3593:
3582:
3581:
3562:
3561:
3542:
3541:
3522:
3454:discovered the
3423:
3410:
3405:
3404:
3375:
3370:
3369:
3348:
3343:
3342:
3320:
3319:
3314:
3303:
3299:
3296:
3268:
3263:
3258:
3257:
3223:
3218:
3217:
3187:
3182:
3181:
3146:
3141:
3140:
3117:
3112:
3107:
3106:
3079:
3047:
3042:
3037:
3036:
3012:
3011:
3001:
3000:
2984:
2982:
2966:
2963:
2962:
2944:
2933:
2931:
2913:
2902:
2895:
2887:
2877:
2874:
2873:
2866:
2865:
2860:
2850:
2842:
2832:
2823:
2822:
2818:recursively by
2798:
2793:
2792:
2767:
2756:
2755:
2734:
2729:
2728:
2649:
2648:
2642:
2634:Walsh functions
2581:
2580:
2549:
2548:
2513:
2508:
2495:
2485:
2484:
2466:
2461:
2456:
2438:
2433:
2430:
2429:
2411:
2406:
2404:
2386:
2381:
2374:
2358:
2353:
2348:
2347:
2327:
2326:
2316:
2315:
2310:
2302:
2294:
2288:
2287:
2279:
2271:
2266:
2260:
2259:
2251:
2246:
2238:
2232:
2231:
2226:
2221:
2216:
2206:
2198:
2188:
2185:
2184:
2174:
2173:
2165:
2159:
2158:
2153:
2143:
2135:
2125:
2122:
2121:
2111:
2110:
2100:
2092:
2082:
2073:
2072:
2040:
2039:
2031:
2025:
2024:
2019:
2009:
2003:
2002:
1984:
1979:
1931:
1930:
1902:
1901:
1863:
1862:
1798:
1797:
1759:
1758:
1716:
1715:
1676:
1675:
1624:
1623:
1598:
1597:
1572:
1571:
1549:
1548:
1529:
1528:
1509:
1508:
1489:
1488:
1458:
1457:
1430:
1429:
1393:
1377:
1331:
1315:
1248:
1232:
1216:
1200:
1154:
1138:
1106:
1105:
1083:
1082:
1063:
1062:
1035:
1019:
1000:
995:
994:
925:
909:
895:
894:
858:
842:
810:
809:
781:
780:
711:
695:
681:
680:
655:
654:
629:
628:
594:
593:
545:
544:
530:
474:
441:
440:
418:
356:
329:
328:
290:
289:
266:identity matrix
261: ×
255:
227:
209:
201:
200:
174:
153:(BRR), used by
109:
104: ×
59:Hadamard matrix
34:
28:
23:
22:
15:
12:
11:
5:
6064:
6062:
6054:
6053:
6048:
6043:
6033:
6032:
6026:
6025:
6023:
6022:
6017:
6012:
5997:
5994:
5993:
5991:
5990:
5985:
5980:
5975:
5973:Perfect matrix
5970:
5965:
5960:
5955:
5949:
5947:
5943:
5942:
5940:
5939:
5934:
5929:
5924:
5919:
5914:
5909:
5904:
5899:
5894:
5889:
5884:
5879:
5873:
5871:
5867:
5866:
5864:
5863:
5858:
5853:
5848:
5843:
5838:
5833:
5828:
5822:
5820:
5813:
5812:
5810:
5809:
5804:
5799:
5794:
5789:
5784:
5779:
5774:
5769:
5764:
5758:
5756:
5749:
5748:
5746:
5745:
5743:Transformation
5740:
5735:
5730:
5725:
5720:
5715:
5710:
5705:
5700:
5695:
5690:
5685:
5680:
5675:
5670:
5665:
5660:
5655:
5650:
5645:
5640:
5635:
5630:
5625:
5620:
5615:
5610:
5605:
5600:
5595:
5589:
5587:
5583:
5582:
5580:
5579:
5574:
5569:
5564:
5559:
5554:
5549:
5544:
5539:
5534:
5529:
5520:
5514:
5512:
5501:
5500:
5498:
5497:
5492:
5487:
5482:
5480:Diagonalizable
5477:
5472:
5467:
5462:
5456:
5454:
5450:Conditions on
5447:
5446:
5444:
5443:
5438:
5433:
5428:
5423:
5418:
5413:
5408:
5403:
5398:
5392:
5390:
5386:
5385:
5383:
5382:
5377:
5372:
5367:
5362:
5357:
5352:
5347:
5342:
5337:
5332:
5330:Skew-symmetric
5327:
5325:Skew-Hermitian
5322:
5317:
5312:
5307:
5302:
5297:
5292:
5287:
5282:
5277:
5272:
5267:
5262:
5257:
5252:
5247:
5242:
5237:
5232:
5227:
5222:
5217:
5212:
5207:
5202:
5197:
5192:
5187:
5182:
5177:
5172:
5167:
5162:
5160:Block-diagonal
5157:
5152:
5147:
5142:
5137:
5135:Anti-symmetric
5132:
5130:Anti-Hermitian
5127:
5122:
5116:
5114:
5110:
5109:
5103:
5101:
5100:
5093:
5086:
5078:
5072:
5071:
5066:
5057:
5051:
5039:
5027:
5019:
5018:External links
5016:
5015:
5014:
5008:
4995:
4968:
4935:
4925:(4): 365–368.
4910:
4898:(2): 188–195.
4883:
4871:(3): 332–338.
4856:
4844:(4): 432–446.
4829:
4811:(1): 235–242.
4798:
4780:(3): 343–344.
4765:
4759:
4746:
4709:
4706:
4704:
4703:
4656:
4625:(4): 929–952.
4603:
4588:
4567:(1): 319–346.
4545:
4532:(3): 332–338.
4512:
4471:
4452:(6): 427–433.
4436:
4407:(6): 270–277.
4391:
4372:(6): 435–440.
4356:
4319:
4298:(3): 237–238.
4276:
4247:
4226:
4174:
4158:
4132:
4130:
4127:
4126:
4125:
4120:
4115:
4110:
4105:
4100:
4093:
4090:
4089:
4088:
4074:
4064:
4058:
4048:
4045:
4038:Coded aperture
4035:
4021:
4013:
4010:
3998:roots of unity
3965:
3940:
3937:
3934:
3923:
3919:
3903:
3900:
3898:less than 10.
3890:
3887:
3835:
3832:
3806:contains 1 if
3778:loses both to
3710:
3707:
3704:
3701:
3698:
3695:
3684:
3663:
3660:
3655:
3652:
3639:
3619:
3616:
3613:
3610:
3606:
3600:
3596:
3592:
3589:
3569:
3549:
3521:
3518:
3430:
3426:
3422:
3417:
3413:
3382:
3378:
3355:
3351:
3315:
3304:
3298:
3295:
3292:
3275:
3271:
3266:
3235:
3230:
3226:
3200:
3197:
3194:
3190:
3153:
3149:
3124:
3120:
3115:
3103:
3102:
3091:
3086:
3082:
3069:
3065:
3061:
3054:
3050:
3045:
3026:
3025:
3010:
3005:
2997:
2994:
2991:
2987:
2983:
2979:
2976:
2973:
2969:
2965:
2964:
2957:
2954:
2951:
2947:
2943:
2940:
2936:
2932:
2926:
2923:
2920:
2916:
2912:
2909:
2905:
2901:
2900:
2898:
2893:
2890:
2888:
2884:
2880:
2876:
2875:
2870:
2864:
2861:
2859:
2856:
2855:
2853:
2848:
2845:
2843:
2839:
2835:
2831:
2830:
2805:
2801:
2774:
2770:
2766:
2763:
2741:
2737:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2641:
2638:
2588:
2568:
2565:
2562:
2559:
2556:
2545:
2544:
2533:
2526:
2523:
2520:
2516:
2511:
2507:
2502:
2498:
2494:
2489:
2479:
2476:
2473:
2469:
2464:
2460:
2457:
2451:
2448:
2445:
2441:
2436:
2432:
2431:
2424:
2421:
2418:
2414:
2409:
2405:
2399:
2396:
2393:
2389:
2384:
2380:
2379:
2377:
2372:
2365:
2361:
2356:
2341:
2340:
2325:
2320:
2314:
2311:
2309:
2306:
2303:
2301:
2298:
2295:
2293:
2290:
2289:
2286:
2283:
2280:
2278:
2275:
2272:
2270:
2267:
2265:
2262:
2261:
2258:
2255:
2252:
2250:
2247:
2245:
2242:
2239:
2237:
2234:
2233:
2230:
2227:
2225:
2222:
2220:
2217:
2215:
2212:
2211:
2209:
2204:
2201:
2199:
2195:
2191:
2187:
2186:
2183:
2178:
2172:
2169:
2166:
2164:
2161:
2160:
2157:
2154:
2152:
2149:
2148:
2146:
2141:
2138:
2136:
2132:
2128:
2124:
2123:
2120:
2115:
2109:
2106:
2105:
2103:
2098:
2095:
2093:
2089:
2085:
2081:
2080:
2066:Walsh matrices
2058:
2057:
2044:
2038:
2035:
2032:
2030:
2027:
2026:
2023:
2020:
2018:
2015:
2014:
2012:
1983:
1980:
1976:
1975:
1958:
1955:
1952:
1949:
1946:
1942:
1938:
1927:
1926:
1915:
1912:
1909:
1899:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1860:
1849:
1846:
1841:
1838:
1833:
1830:
1827:
1824:
1821:
1816:
1813:
1808:
1805:
1795:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1743:
1740:
1737:
1733:
1729:
1726:
1723:
1703:
1700:
1697:
1693:
1689:
1686:
1683:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1635:
1631:
1611:
1608:
1605:
1585:
1582:
1579:
1556:
1536:
1516:
1496:
1473:
1469:
1465:
1445:
1441:
1437:
1426:
1425:
1414:
1411:
1406:
1403:
1400:
1396:
1390:
1387:
1384:
1380:
1374:
1371:
1368:
1363:
1360:
1357:
1353:
1349:
1344:
1341:
1338:
1334:
1328:
1325:
1322:
1318:
1312:
1307:
1304:
1301:
1297:
1291:
1288:
1285:
1280:
1277:
1274:
1270:
1266:
1261:
1258:
1255:
1251:
1245:
1242:
1239:
1235:
1229:
1226:
1223:
1219:
1213:
1210:
1207:
1203:
1197:
1194:
1191:
1186:
1183:
1180:
1176:
1172:
1167:
1164:
1161:
1157:
1151:
1148:
1145:
1141:
1135:
1132:
1129:
1124:
1121:
1118:
1114:
1090:
1070:
1048:
1045:
1042:
1038:
1032:
1029:
1026:
1022:
1018:
1013:
1010:
1007:
1003:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
928:
922:
919:
916:
912:
908:
905:
902:
891:
890:
879:
876:
871:
868:
865:
861:
855:
852:
849:
845:
839:
836:
833:
828:
825:
822:
818:
794:
791:
788:
766:
763:
760:
757:
754:
751:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
718:
714:
708:
705:
702:
698:
694:
691:
688:
668:
665:
662:
642:
639:
636:
616:
613:
610:
607:
604:
601:
558:
555:
552:
532:
531:
528:
523:
513:if and only if
506:
505:
494:
489:
485:
481:
477:
473:
469:
465:
462:
459:
456:
453:
449:
428:between 1 and
416:
388:
387:
376:
371:
367:
363:
359:
354:
351:
348:
345:
342:
339:
336:
305:
299:
253:
248:
247:
234:
230:
226:
223:
212:
208:
173:
170:
120:absolute value
43:Gilbert Strang
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6063:
6052:
6049:
6047:
6044:
6042:
6039:
6038:
6036:
6021:
6018:
6016:
6013:
6011:
6010:
6005:
5999:
5998:
5995:
5989:
5986:
5984:
5981:
5979:
5978:Pseudoinverse
5976:
5974:
5971:
5969:
5966:
5964:
5961:
5959:
5956:
5954:
5951:
5950:
5948:
5946:Related terms
5944:
5938:
5937:Z (chemistry)
5935:
5933:
5930:
5928:
5925:
5923:
5920:
5918:
5915:
5913:
5910:
5908:
5905:
5903:
5900:
5898:
5895:
5893:
5890:
5888:
5885:
5883:
5880:
5878:
5875:
5874:
5872:
5868:
5862:
5859:
5857:
5854:
5852:
5849:
5847:
5844:
5842:
5839:
5837:
5834:
5832:
5829:
5827:
5824:
5823:
5821:
5819:
5814:
5808:
5805:
5803:
5800:
5798:
5795:
5793:
5790:
5788:
5785:
5783:
5780:
5778:
5775:
5773:
5770:
5768:
5765:
5763:
5760:
5759:
5757:
5755:
5750:
5744:
5741:
5739:
5736:
5734:
5731:
5729:
5726:
5724:
5721:
5719:
5716:
5714:
5711:
5709:
5706:
5704:
5701:
5699:
5696:
5694:
5691:
5689:
5686:
5684:
5681:
5679:
5676:
5674:
5671:
5669:
5666:
5664:
5661:
5659:
5656:
5654:
5651:
5649:
5646:
5644:
5641:
5639:
5636:
5634:
5631:
5629:
5626:
5624:
5621:
5619:
5616:
5614:
5611:
5609:
5606:
5604:
5601:
5599:
5596:
5594:
5591:
5590:
5588:
5584:
5578:
5575:
5573:
5570:
5568:
5565:
5563:
5560:
5558:
5555:
5553:
5550:
5548:
5545:
5543:
5540:
5538:
5535:
5533:
5530:
5528:
5524:
5521:
5519:
5516:
5515:
5513:
5511:
5507:
5502:
5496:
5493:
5491:
5488:
5486:
5483:
5481:
5478:
5476:
5473:
5471:
5468:
5466:
5463:
5461:
5458:
5457:
5455:
5453:
5448:
5442:
5439:
5437:
5434:
5432:
5429:
5427:
5424:
5422:
5419:
5417:
5414:
5412:
5409:
5407:
5404:
5402:
5399:
5397:
5394:
5393:
5391:
5387:
5381:
5378:
5376:
5373:
5371:
5368:
5366:
5363:
5361:
5358:
5356:
5353:
5351:
5348:
5346:
5343:
5341:
5338:
5336:
5333:
5331:
5328:
5326:
5323:
5321:
5318:
5316:
5313:
5311:
5308:
5306:
5303:
5301:
5298:
5296:
5295:Pentadiagonal
5293:
5291:
5288:
5286:
5283:
5281:
5278:
5276:
5273:
5271:
5268:
5266:
5263:
5261:
5258:
5256:
5253:
5251:
5248:
5246:
5243:
5241:
5238:
5236:
5233:
5231:
5228:
5226:
5223:
5221:
5218:
5216:
5213:
5211:
5208:
5206:
5203:
5201:
5198:
5196:
5193:
5191:
5188:
5186:
5183:
5181:
5178:
5176:
5173:
5171:
5168:
5166:
5163:
5161:
5158:
5156:
5153:
5151:
5148:
5146:
5143:
5141:
5138:
5136:
5133:
5131:
5128:
5126:
5125:Anti-diagonal
5123:
5121:
5118:
5117:
5115:
5111:
5106:
5099:
5094:
5092:
5087:
5085:
5080:
5079:
5076:
5070:
5067:
5065:
5061:
5058:
5055:
5052:
5048:
5044:
5040:
5038:
5032:
5028:
5025:
5022:
5021:
5017:
5011:
5005:
5001:
4996:
4991:
4986:
4982:
4978:
4977:Discrete Math
4974:
4969:
4965:
4961:
4957:
4953:
4949:
4945:
4941:
4936:
4932:
4928:
4924:
4920:
4916:
4911:
4906:
4901:
4897:
4893:
4889:
4884:
4879:
4874:
4870:
4866:
4862:
4857:
4852:
4847:
4843:
4839:
4835:
4830:
4826:
4822:
4818:
4814:
4810:
4806:
4805:
4799:
4795:
4791:
4787:
4783:
4779:
4775:
4771:
4766:
4762:
4756:
4752:
4747:
4743:
4739:
4734:
4729:
4725:
4721:
4717:
4712:
4711:
4707:
4699:
4695:
4691:
4687:
4683:
4679:
4675:
4671:
4667:
4660:
4657:
4652:
4648:
4643:
4638:
4633:
4628:
4624:
4620:
4619:
4614:
4607:
4604:
4599:
4592:
4589:
4584:
4580:
4575:
4570:
4566:
4562:
4561:
4556:
4549:
4546:
4540:
4535:
4531:
4527:
4523:
4516:
4513:
4508:
4504:
4499:
4494:
4490:
4486:
4482:
4475:
4472:
4467:
4463:
4459:
4455:
4451:
4447:
4440:
4437:
4432:
4428:
4424:
4420:
4415:
4410:
4406:
4402:
4395:
4392:
4387:
4383:
4379:
4375:
4371:
4367:
4360:
4357:
4352:
4348:
4344:
4340:
4336:
4332:
4331:
4323:
4320:
4315:
4311:
4306:
4301:
4297:
4293:
4292:
4287:
4280:
4277:
4272:
4268:
4264:
4260:
4259:
4251:
4248:
4243:
4239:
4238:
4230:
4227:
4221:
4217:
4213:
4209:
4204:
4199:
4195:
4191:
4190:
4185:
4178:
4175:
4171:
4168:
4162:
4159:
4143:
4137:
4134:
4128:
4124:
4121:
4119:
4116:
4114:
4111:
4109:
4106:
4104:
4101:
4099:
4096:
4095:
4091:
4086:
4082:
4078:
4075:
4072:
4068:
4065:
4062:
4059:
4056:
4052:
4049:
4046:
4043:
4039:
4036:
4033:
4029:
4025:
4022:
4019:
4016:
4015:
4011:
4009:
4007:
4003:
3999:
3996:
3992:
3988:
3984:
3980:
3976:
3972:
3968:
3961:
3957:
3952:
3938:
3935:
3932:
3921:
3917:
3909:
3901:
3899:
3897:
3888:
3886:
3884:
3880:
3876:
3872:
3868:
3864:
3860:
3856:
3855:square number
3852:
3848:
3844:
3840:
3833:
3831:
3829:
3825:
3821:
3817:
3813:
3810: =
3809:
3805:
3801:
3797:
3793:
3789:
3785:
3781:
3777:
3773:
3769:
3765:
3761:
3757:
3753:
3749:
3744:
3740:
3736:
3732:
3728:
3723:
3708:
3705:
3702:
3699:
3696:
3693:
3682:
3673:
3669:
3661:
3659:
3654:Special cases
3653:
3651:
3637:
3614:
3611:
3608:
3604:
3598:
3594:
3587:
3567:
3547:
3538:
3536:
3535:transposition
3532:
3527:
3519:
3517:
3514:
3510:
3508:
3504:
3500:
3496:
3492:
3487:
3485:
3484:finite fields
3481:
3477:
3473:
3469:
3465:
3461:
3457:
3453:
3452:Raymond Paley
3448:
3446:
3428:
3424:
3420:
3415:
3411:
3402:
3398:
3380:
3376:
3353:
3349:
3339:
3337:
3333:
3329:
3325:
3324:open question
3318:
3312:
3308:
3293:
3291:
3273:
3269:
3264:
3255:
3254:Hadamard code
3251:
3246:
3233:
3228:
3224:
3216:
3198:
3195:
3192:
3188:
3180:
3176:
3173:
3169:
3151:
3147:
3122:
3118:
3113:
3089:
3084:
3080:
3067:
3063:
3059:
3052:
3048:
3043:
3035:
3034:
3033:
3031:
3008:
3003:
2995:
2992:
2989:
2985:
2977:
2974:
2971:
2967:
2955:
2952:
2949:
2945:
2941:
2938:
2934:
2924:
2921:
2918:
2914:
2910:
2907:
2903:
2896:
2891:
2889:
2882:
2878:
2868:
2862:
2857:
2851:
2846:
2844:
2837:
2833:
2821:
2820:
2819:
2803:
2799:
2790:
2772:
2768:
2764:
2761:
2739:
2735:
2711:
2708:
2699:
2696:
2693:
2678:
2675:
2669:
2666:
2663:
2660:
2647:
2639:
2637:
2635:
2631:
2627:
2623:
2619:
2614:
2612:
2609:
2604:
2602:
2586:
2566:
2563:
2560:
2557:
2554:
2531:
2524:
2521:
2518:
2514:
2509:
2505:
2500:
2496:
2492:
2487:
2477:
2474:
2471:
2467:
2462:
2458:
2449:
2446:
2443:
2439:
2434:
2422:
2419:
2416:
2412:
2407:
2397:
2394:
2391:
2387:
2382:
2375:
2370:
2363:
2359:
2354:
2346:
2345:
2344:
2323:
2318:
2312:
2307:
2304:
2299:
2296:
2291:
2284:
2281:
2276:
2273:
2268:
2263:
2256:
2253:
2248:
2243:
2240:
2235:
2228:
2223:
2218:
2213:
2207:
2202:
2200:
2193:
2189:
2181:
2176:
2170:
2167:
2162:
2155:
2150:
2144:
2139:
2137:
2130:
2126:
2118:
2113:
2107:
2101:
2096:
2094:
2087:
2083:
2071:
2070:
2069:
2067:
2063:
2042:
2036:
2033:
2028:
2021:
2016:
2010:
2001:
2000:
1999:
1997:
1993:
1990:in 1867. Let
1989:
1974:
1972:
1971:contradiction
1956:
1953:
1950:
1947:
1944:
1940:
1936:
1913:
1910:
1907:
1900:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1861:
1847:
1844:
1839:
1836:
1831:
1828:
1825:
1822:
1819:
1814:
1811:
1806:
1803:
1796:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1757:
1756:
1755:
1741:
1738:
1735:
1731:
1727:
1724:
1721:
1701:
1698:
1695:
1691:
1687:
1684:
1681:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1633:
1629:
1609:
1606:
1603:
1583:
1580:
1577:
1568:
1554:
1534:
1514:
1494:
1485:
1471:
1467:
1463:
1443:
1439:
1435:
1412:
1409:
1404:
1401:
1398:
1394:
1388:
1385:
1382:
1378:
1372:
1369:
1366:
1361:
1358:
1355:
1351:
1347:
1342:
1339:
1336:
1332:
1326:
1323:
1320:
1316:
1310:
1305:
1302:
1299:
1295:
1289:
1286:
1283:
1278:
1275:
1272:
1268:
1264:
1259:
1256:
1253:
1249:
1243:
1240:
1237:
1233:
1227:
1224:
1221:
1217:
1211:
1208:
1205:
1201:
1195:
1192:
1189:
1184:
1181:
1178:
1174:
1170:
1165:
1162:
1159:
1155:
1149:
1146:
1143:
1139:
1133:
1130:
1127:
1122:
1119:
1116:
1112:
1104:
1103:
1102:
1088:
1068:
1046:
1043:
1040:
1036:
1030:
1027:
1024:
1020:
1016:
1011:
1008:
1005:
1001:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
945:
939:
936:
933:
930:
920:
917:
914:
910:
903:
900:
877:
874:
869:
866:
863:
859:
853:
850:
847:
843:
837:
834:
831:
826:
823:
820:
816:
808:
807:
806:
792:
789:
786:
761:
758:
755:
752:
749:
746:
743:
740:
737:
734:
731:
725:
722:
719:
716:
706:
703:
700:
696:
689:
686:
666:
663:
660:
640:
637:
634:
614:
611:
608:
605:
602:
599:
590:
588:
584:
580:
576:
572:
556:
553:
550:
541:
539:
534:
533:
527:
526:
522:
519:
517:
514:
511:
492:
487:
483:
479:
475:
471:
460:
454:
451:
439:
438:
437:
435:
431:
427:
423:
419:
412:
408:
404:
401:Suppose that
399:
397:
393:
374:
369:
365:
361:
357:
352:
349:
343:
337:
334:
327:
326:
325:
323:
319:
303:
297:
287:
283:
279:
275:
271:
267:
264:
260:
256:
232:
228:
224:
221:
210:
206:
199:
198:
197:
195:
191:
187:
183:
179:
171:
169:
167:
164:
160:
156:
155:statisticians
152:
148:
144:
143:Hadamard code
140:
135:
133:
129:
125:
121:
117:
112:
107:
103:
99:
98:parallelotope
96:-dimensional
95:
90:
87:
86:combinatorial
83:
80:
79:perpendicular
76:
72:
68:
67:square matrix
64:
60:
56:
48:
44:
40:
36:
33:
19:
6000:
5932:Substitution
5818:graph theory
5315:Quaternionic
5305:Persymmetric
5239:
4999:
4980:
4976:
4947:
4943:
4922:
4918:
4895:
4891:
4868:
4864:
4841:
4837:
4808:
4802:
4777:
4773:
4750:
4723:
4719:
4673:
4669:
4659:
4622:
4616:
4606:
4597:
4591:
4564:
4558:
4548:
4529:
4525:
4515:
4488:
4484:
4474:
4449:
4445:
4439:
4404:
4400:
4394:
4369:
4365:
4359:
4337:(1): 65–81.
4334:
4328:
4322:
4295:
4289:
4279:
4262:
4256:
4250:
4241:
4235:
4229:
4193:
4187:
4177:
4166:
4161:
4149:. Retrieved
4136:
4113:Walsh matrix
4005:
4001:
3994:
3978:
3970:
3963:
3953:
3905:
3895:
3892:
3882:
3878:
3874:
3873:> 1 then
3870:
3866:
3862:
3850:
3846:
3842:
3837:
3827:
3823:
3819:
3815:
3811:
3807:
3803:
3799:
3795:
3791:
3787:
3783:
3779:
3775:
3771:
3767:
3763:
3759:
3755:
3751:
3747:
3742:
3738:
3734:
3730:
3724:
3671:
3667:
3665:
3657:
3539:
3523:
3515:
3511:
3488:
3479:
3475:
3463:
3459:
3449:
3444:
3400:
3396:
3340:
3335:
3331:
3327:
3321:
3310:
3306:
3247:
3174:
3104:
3027:
2788:
2643:
2621:
2615:
2610:
2605:
2599:denotes the
2546:
2342:
2061:
2059:
1995:
1991:
1985:
1928:
1569:
1486:
1427:
1061:. Note that
892:
591:
582:
578:
570:
542:
535:
520:
515:
509:
507:
436:states that
429:
425:
421:
414:
410:
402:
400:
395:
391:
389:
317:
286:real numbers
277:
273:
269:
262:
258:
251:
249:
189:
181:
177:
175:
136:
113:-dimensional
110:
105:
101:
93:
91:
58:
52:
35:
32:Walsh matrix
5907:Hamiltonian
5831:Biadjacency
5767:Correlation
5683:Householder
5633:Commutation
5370:Vandermonde
5365:Tridiagonal
5300:Permutation
5290:Nonnegative
5275:Matrix unit
5155:Bisymmetric
4642:10356/92085
4151:11 February
4147:. UC Denver
4018:Olivia MFSK
4000:. The term
3501:in 1962 at
3468:prime power
993:by setting
196:. In fact:
124:determinant
84:, while in
55:mathematics
6035:Categories
5807:Transition
5802:Stochastic
5772:Covariance
5754:statistics
5733:Symplectic
5728:Similarity
5557:Unimodular
5552:Orthogonal
5537:Involutory
5532:Invertible
5527:Projection
5523:Idempotent
5465:Convergent
5360:Triangular
5310:Polynomial
5255:Hessenberg
5225:Equivalent
5220:Elementary
5200:Copositive
5190:Conference
5150:Bidiagonal
4720:Math. Comp
4244:: 240–246.
3822:and −1 if
3741:, each of
3727:tournament
3526:equivalent
3507:Williamson
3478:+ 1) when
3250:Walsh code
2620:and, when
390:where det(
172:Properties
71:orthogonal
30:See also:
5988:Wronskian
5912:Irregular
5902:Gell-Mann
5851:Laplacian
5846:Incidence
5826:Adjacency
5797:Precision
5762:Centering
5668:Generator
5638:Confusion
5623:Circulant
5603:Augmented
5562:Unipotent
5542:Nilpotent
5518:Congruent
5495:Stieltjes
5470:Defective
5460:Companion
5431:Redheffer
5350:Symmetric
5345:Sylvester
5320:Signature
5250:Hermitian
5230:Frobenius
5140:Arrowhead
5120:Alternant
5060:R-Package
4698:122560830
4690:0004-9727
4507:126730552
4491:: 33–46.
4466:121547091
4414:1301.3671
3964:H H = n I
3859:circulant
3802:, column
3729:of order
3612:
3560:of order
3472:congruent
3462:+ 1 when
3421:⊗
3196:−
3030:induction
2993:−
2975:−
2953:−
2942:×
2922:−
2911:×
2765:×
2712:⊕
2685:↦
2679:×
2667:−
2618:symmetric
2587:⊗
2564:∈
2558:≤
2522:−
2506:⊗
2475:−
2459:−
2447:−
2420:−
2395:−
2305:−
2297:−
2282:−
2274:−
2254:−
2241:−
2168:−
2034:−
1957:β
1951:α
1914:β
1908:α
1887:α
1884:−
1881:β
1875:β
1872:−
1869:α
1848:α
1845:−
1829:β
1823:β
1820:−
1804:α
1783:γ
1777:β
1771:δ
1765:α
1742:β
1739:−
1722:δ
1702:α
1699:−
1682:γ
1662:δ
1656:β
1650:γ
1644:α
1610:γ
1604:β
1584:δ
1578:α
1555:δ
1535:γ
1515:β
1495:α
1370:−
1352:∑
1287:−
1269:∑
1193:−
1175:∑
1131:−
1113:∑
973:−
940:∈
835:−
817:∑
790:≠
759:−
726:∈
664:×
638:≥
472:≤
455:
353:±
338:
316:Dividing
186:transpose
166:estimator
163:parameter
75:geometric
6046:Matrices
5816:Used in
5752:Used in
5713:Rotation
5688:Jacobian
5648:Distance
5628:Cofactor
5613:Carleman
5593:Adjugate
5577:Weighing
5510:inverses
5506:products
5475:Definite
5406:Identity
5396:Exchange
5389:Constant
5355:Toeplitz
5240:Hadamard
5210:Diagonal
4825:39169723
4794:14193297
4431:26598685
4386:17206302
4092:See also
4083:and the
4028:variance
3826:defeats
3818:defeats
3470:that is
2579:, where
585:must be
577:for odd
159:variance
141:using a
128:matrices
5917:Overlap
5882:Density
5841:Edmonds
5718:Seifert
5678:Hessian
5643:Coxeter
5567:Unitary
5485:Hurwitz
5416:Of ones
5401:Hilbert
5335:Skyline
5280:Metzler
5270:Logical
5265:Integer
5175:Boolean
5107:classes
4944:Metrika
4742:0179093
4651:2646093
4583:0179098
4351:0009590
4314:0148686
4220:0523759
4212:2958712
3973:is the
3960:modulus
3531:coarser
3491:Baumert
3466:is any
3166:linear
2608:integer
432:. Then
407:complex
257:is the
194:inverse
82:vectors
65:, is a
5836:Degree
5777:Design
5708:Random
5698:Payoff
5693:Moment
5618:Cartan
5608:Bézout
5547:Normal
5421:Pascal
5411:Lehmer
5340:Sparse
5260:Hollow
5245:Hankel
5180:Cauchy
5105:Matrix
5006:
4962:
4823:
4792:
4757:
4740:
4696:
4688:
4649:
4581:
4505:
4464:
4429:
4384:
4349:
4312:
4218:
4210:
3969:where
3497:, and
3495:Golomb
3252:. The
3177:, and
2754:, the
250:where
184:. The
126:among
116:volume
5897:Gamma
5861:Tutte
5723:Shear
5436:Shift
5426:Pauli
5375:Walsh
5285:Moore
5165:Block
4964:40646
4960:S2CID
4821:S2CID
4790:S2CID
4694:S2CID
4647:JSTOR
4503:S2CID
4462:S2CID
4427:S2CID
4409:arXiv
4382:S2CID
4208:JSTOR
4145:(PDF)
4129:Notes
4030:of a
4008:= 4.
3885:odd.
3881:with
3853:be a
3786:− 1)(
3213:with
2626:trace
627:with
581:, so
538:proof
529:Proof
405:is a
282:field
161:of a
73:. In
5703:Pick
5673:Gram
5441:Zero
5145:Band
5037:OEIS
5004:ISBN
4755:ISBN
4686:ISSN
4153:2023
4079:for
4069:for
3857:. A
3672:skew
3580:has
3499:Hall
3399:and
3368:and
3172:rank
2547:for
2343:and
1714:and
1487:Let
587:even
554:>
536:The
268:and
176:Let
92:The
57:, a
5792:Hat
5525:or
5508:or
5035:in
4985:doi
4981:140
4952:doi
4927:doi
4900:doi
4873:doi
4846:doi
4813:doi
4782:doi
4728:doi
4678:doi
4637:hdl
4627:doi
4569:doi
4534:doi
4493:doi
4489:778
4454:doi
4419:doi
4374:doi
4339:doi
4300:doi
4267:doi
4198:doi
3977:of
3814:or
3674:if
3670:is
3609:log
3503:JPL
3170:of
592:If
575:odd
543:If
452:det
335:det
284:of
188:of
53:In
47:MIT
6037::
5045:.
4979:.
4975:.
4958:.
4948:62
4946:.
4942:.
4921:.
4917:.
4896:21
4894:.
4890:.
4869:12
4867:.
4863:.
4842:17
4840:.
4836:.
4819:.
4807:.
4788:.
4778:11
4776:.
4772:.
4738:MR
4736:.
4724:19
4722:.
4718:.
4692:.
4684:.
4674:10
4672:.
4668:.
4645:.
4635:.
4623:12
4621:.
4615:.
4579:MR
4577:.
4565:15
4563:.
4557:.
4530:12
4528:.
4524:.
4501:.
4487:.
4483:.
4460:.
4450:53
4448:.
4425:.
4417:.
4405:22
4403:.
4380:.
4370:13
4368:.
4347:MR
4345:.
4335:11
4333:.
4310:MR
4308:.
4296:68
4294:.
4288:.
4263:12
4261:.
4242:17
4240:.
4216:MR
4214:.
4206:.
4192:.
4186:.
3989:.
3770:,
3493:,
3486:.
3445:nm
2636:.
2613:.
2603:.
2068:.
1973:.
1969:,
1413:0.
805::
589:.
424:,
417:ij
398:.
168:.
134:.
5922:S
5380:Z
5097:e
5090:t
5083:v
5064:R
5049:.
5033:.
5012:.
4993:.
4987::
4966:.
4954::
4933:.
4929::
4923:4
4908:.
4902::
4881:.
4875::
4854:.
4848::
4827:.
4815::
4809:5
4796:.
4784::
4763:.
4744:.
4730::
4700:.
4680::
4653:.
4639::
4629::
4585:.
4571::
4542:.
4536::
4509:.
4495::
4468:.
4456::
4433:.
4421::
4411::
4388:.
4376::
4353:.
4341::
4316:.
4302::
4273:.
4269::
4224:.
4222:.
4200::
4194:6
4155:.
4057:.
4034:.
4006:q
3995:q
3979:H
3971:H
3966:n
3939:I
3936:w
3933:=
3927:T
3922:W
3918:W
3896:u
3883:u
3879:u
3875:n
3871:n
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3750:(
3748:n
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3706:I
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3697:H
3694:+
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3668:H
3638:H
3618:)
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3605:/
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3591:(
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3397:n
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3377:H
3354:n
3350:H
3336:k
3332:k
3313:?
3311:k
3307:k
3302::
3274:n
3270:2
3265:H
3234:.
3229:n
3225:F
3199:1
3193:n
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3175:n
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3148:2
3123:n
3119:2
3114:H
3090:.
3085:n
3081:F
3074:T
3068:n
3064:F
3060:=
3053:n
3049:2
3044:H
3009:.
3004:]
2996:1
2990:n
2986:F
2978:1
2972:n
2968:F
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2950:n
2946:2
2939:1
2935:1
2925:1
2919:n
2915:2
2908:1
2904:0
2897:[
2892:=
2883:n
2879:F
2869:]
2863:1
2858:0
2852:[
2847:=
2838:1
2834:F
2804:n
2800:F
2789:n
2773:n
2769:2
2762:n
2740:n
2736:F
2715:}
2709:,
2706:)
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2700:1
2697:,
2694:0
2691:{
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2682:)
2676:,
2673:}
2670:1
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2661:1
2658:{
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2493:=
2488:]
2478:1
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2468:2
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2450:1
2444:k
2440:2
2435:H
2423:1
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2413:2
2408:H
2398:1
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2383:H
2376:[
2371:=
2364:k
2360:2
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2324:,
2319:]
2313:1
2308:1
2300:1
2292:1
2285:1
2277:1
2269:1
2264:1
2257:1
2249:1
2244:1
2236:1
2229:1
2224:1
2219:1
2214:1
2208:[
2203:=
2194:4
2190:H
2182:,
2177:]
2171:1
2163:1
2156:1
2151:1
2145:[
2140:=
2131:2
2127:H
2119:,
2114:]
2108:1
2102:[
2097:=
2088:1
2084:H
2062:n
2043:]
2037:H
2029:H
2022:H
2017:H
2011:[
1996:n
1992:H
1954:+
1948:=
1945:2
1941:/
1937:n
1911:=
1878:=
1840:2
1837:n
1832:+
1826:=
1815:2
1812:n
1807:+
1780:+
1774:=
1768:+
1736:2
1732:/
1728:n
1725:=
1696:2
1692:/
1688:n
1685:=
1659:+
1653:=
1647:+
1641:=
1638:2
1634:/
1630:n
1607:+
1581:+
1472:2
1468:/
1464:n
1444:2
1440:/
1436:n
1410:=
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1386:,
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1373:1
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1348:=
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1273:i
1265:=
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1120:=
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904:=
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744:.
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690:=
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667:n
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600:n
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484:/
480:n
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350:=
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298:n
278:H
274:H
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254:n
252:I
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229:I
225:n
222:=
216:T
211:H
207:H
190:H
182:n
178:H
111:n
106:n
102:n
94:n
20:)
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