813:
1284:
5052:
1070:
1279:{\displaystyle {\begin{aligned}&&\mathbf {A} ^{\textsf {T}}\mathbf {b} &-\mathbf {A} ^{\textsf {T}}\mathbf {Ax} =0\\\Rightarrow &&\mathbf {A} ^{\textsf {T}}\mathbf {b} &=\mathbf {A} ^{\textsf {T}}\mathbf {Ax} \\\Rightarrow &&\mathbf {x} &=\left(\mathbf {A} ^{\textsf {T}}\mathbf {A} \right)^{-1}\mathbf {A} ^{\textsf {T}}\mathbf {b} \end{aligned}}}
2095:
1845:
2215:
657:
452:
1719:
3049:
1937:
3384:
3642:
1971:
1730:
1476:
801:
2375:
2107:
1533:
3290:
2488:
568:
3545:
2960:
357:
1058:
2434:
1631:
232:
2967:
731:
3444:
2549:
519:
3702:
2747:
2707:
2090:{\displaystyle {\hat {\mathbf {\beta } }}_{\text{GLS}}=\left(\mathbf {X} ^{\textsf {T}}\mathbf {\Sigma } ^{-1}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}\mathbf {\Sigma } ^{-1}\mathbf {y} }
1856:
3295:
1075:
3553:
3125:
2610:
1840:{\displaystyle {\hat {\mathbf {y} }}=\mathbf {X} {\hat {\boldsymbol {\beta }}}=\mathbf {X} \left(\mathbf {X} ^{\textsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}\mathbf {y} .}
3808:
The hat matrix was introduced by John Wilder in 1972. An article by
Hoaglin, D.C. and Welsch, R.E. (1978) gives the properties of the matrix and also many examples of its application.
2907:
1406:
264:
184:
683:
2263:
3879:
105:
70:
1313:
974:
3798:
3776:
3750:
3724:
3237:
3188:
3166:
3072:
2871:
2849:
2811:
2775:
2657:
2632:
2573:
2511:
2305:
1558:
1401:
1379:
1357:
1335:
996:
949:
927:
902:
880:
858:
836:
556:
477:
349:
316:
286:
155:
4710:
2210:{\displaystyle \mathbf {H} =\mathbf {X} \left(\mathbf {X} ^{\textsf {T}}\mathbf {\Sigma } ^{-1}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}\mathbf {\Sigma } ^{-1}}
739:
2310:
1492:
3242:
652:{\displaystyle \mathbf {\Sigma } _{\mathbf {r} }=\left(\mathbf {I} -\mathbf {P} \right)^{\textsf {T}}\mathbf {\Sigma } \left(\mathbf {I} -\mathbf {P} \right)}
2439:
4924:
3452:
447:{\displaystyle \mathbf {r} =\mathbf {y} -\mathbf {\hat {y}} =\mathbf {y} -\mathbf {P} \mathbf {y} =\left(\mathbf {I} -\mathbf {P} \right)\mathbf {y} .}
2912:
4143:
5015:
690:
3168:, which is the number of independent parameters of the linear model. For other models such as LOESS that are still linear in the observations
1004:
4109:
1714:{\displaystyle {\hat {\boldsymbol {\beta }}}=\left(\mathbf {X} ^{\textsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}\mathbf {y} ,}
4934:
4700:
2387:
1622:
1584:
327:
4058:
4025:
4000:
3863:
3726:
is a column of all ones, which allows one to analyze the effects of adding an intercept term to a regression. Another use is in the
812:
3044:{\displaystyle \left(\mathbf {I} -\mathbf {P} \right)\mathbf {P} =\mathbf {P} \left(\mathbf {I} -\mathbf {P} \right)=\mathbf {0} .}
192:
3886:
1957:
The above may be generalized to the cases where the weights are not identical and/or the errors are correlated. Suppose that the
2378:
696:
4735:
3827:
3191:
116:
4282:
3389:
2516:
1932:{\displaystyle \mathbf {P} :=\mathbf {X} \left(\mathbf {X} ^{\textsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}.}
486:
3647:
3379:{\displaystyle \mathbf {P} :=\mathbf {X} \left(\mathbf {X} ^{\textsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}}
3637:{\displaystyle \mathbf {P} =\mathbf {A} \left(\mathbf {A} ^{\textsf {T}}\mathbf {A} \right)^{-1}\mathbf {A} ^{\textsf {T}}}
2712:
2672:
5088:
4499:
4136:
4574:
3832:
3817:
31:
4730:
4252:
3973:
3093:
2578:
4834:
4705:
4619:
3915:
2876:
1952:
4939:
4829:
4537:
4217:
998:. A vector that is orthogonal to the column space of a matrix is in the nullspace of the matrix transpose, so
1471:{\displaystyle \mathbf {A} \left(\mathbf {A} ^{\textsf {T}}\mathbf {A} \right)^{-1}\mathbf {A} ^{\textsf {T}}}
4974:
4903:
4785:
4645:
4242:
4129:
3206:
3756:
of the dummy variables for the fixed effect terms. One can use this partition to compute the hat matrix of
4844:
4427:
4232:
3139:
1948:
1616:
240:
160:
4790:
4527:
4377:
4372:
4207:
4182:
4177:
3822:
3198:
3143:
2278:
1486:
Suppose that we wish to estimate a linear model using linear least squares. The model can be written as
689:
of the error vector (and by extension, the response vector as well). For the case of linear models with
666:
120:
5051:
3907:
2223:
123:, which describe the influence each response value has on the fitted value for that same observation.
4984:
4342:
4172:
4152:
1561:
119:
each response value has on each fitted value. The diagonal elements of the projection matrix are the
82:
47:
5005:
4979:
4557:
4362:
4352:
3727:
1293:
954:
796:{\displaystyle \mathbf {\Sigma } _{\mathbf {r} }=\left(\mathbf {I} -\mathbf {P} \right)\sigma ^{2}}
3781:
3759:
3733:
3707:
3220:
3171:
3149:
3055:
2854:
2832:
2794:
2758:
2640:
2615:
2556:
2494:
2370:{\displaystyle \left(\mathbf {X} ^{\textsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}}
2288:
1541:
1384:
1362:
1340:
1318:
979:
932:
910:
885:
863:
841:
819:
539:
460:
332:
299:
269:
138:
5056:
5010:
5000:
4954:
4949:
4878:
4814:
4680:
4417:
4412:
4347:
4337:
4202:
3942:
1592:
3202:
4101:
5093:
5067:
4854:
4849:
4819:
4780:
4775:
4604:
4599:
4584:
4579:
4570:
4565:
4512:
4407:
4357:
4302:
4272:
4267:
4247:
4237:
4197:
4105:
4054:
4050:
4021:
3996:
3965:
3859:
3853:
3087:
1958:
1600:
1588:
686:
559:
533:
132:
108:
1528:{\displaystyle \mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }},}
5062:
5030:
4959:
4898:
4893:
4873:
4809:
4715:
4685:
4670:
4655:
4650:
4589:
4542:
4517:
4507:
4478:
4397:
4392:
4367:
4297:
4277:
4187:
4167:
4093:
3932:
3924:
3128:
3083:
1596:
3704:. There are a number of applications of such a decomposition. In the classical application
4760:
4695:
4675:
4660:
4640:
4624:
4522:
4453:
4443:
4402:
4287:
4257:
480:
3285:{\displaystyle \mathbf {X} ={\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}}
1583:
Many types of models and techniques are subject to this formulation. A few examples are
5020:
4964:
4944:
4929:
4888:
4765:
4725:
4690:
4614:
4553:
4532:
4473:
4463:
4448:
4382:
4327:
4317:
4312:
4222:
3989:
2274:
2273:
The projection matrix has a number of useful algebraic properties. In the language of
5082:
5025:
4883:
4824:
4755:
4745:
4740:
4665:
4594:
4468:
4458:
4387:
4307:
4292:
4227:
4094:
4043:
3753:
3135:
2483:{\displaystyle \mathbf {u} =\mathbf {y} -\mathbf {P} \mathbf {y} \perp \mathbf {X} .}
1604:
1565:
4908:
4865:
4770:
4483:
4422:
4332:
4212:
3880:"Data Assimilation: Observation influence diagnostic of a data assimilation system"
3079:
2282:
112:
2381:.) Some facts of the projection matrix in this setting are summarized as follows:
3540:{\displaystyle \mathbf {P} =\mathbf {P} +\mathbf {P} {\big \mathbf {B} {\big ]},}
4750:
4720:
4488:
4322:
4192:
3197:
Practical applications of the projection matrix in regression analysis include
2955:{\displaystyle \left(\mathbf {I} -\mathbf {P} \right)\mathbf {X} =\mathbf {0} }
838:
has its column space depicted as the green line. The projection of some vector
17:
4801:
4262:
3209:, i.e. observations which have a large effect on the results of a regression.
3131:, for example, the hat matrix is in general neither symmetric nor idempotent.
2752:
293:
38:
4092:
Rao, C. Radhakrishna; Toutenburg, Helge; Shalabh; Heumann, Christian (2008).
5035:
4609:
4074:
4969:
1053:{\displaystyle \mathbf {A} ^{\textsf {T}}(\mathbf {b} -\mathbf {Ax} )=0}
976:, and is one where we can draw a line orthogonal to the column space of
3946:
3937:
2429:{\displaystyle \mathbf {u} =(\mathbf {I} -\mathbf {P} )\mathbf {y} ,}
4075:"Proof that trace of 'hat' matrix in linear regression is rank of X"
3928:
907:
From the figure, it is clear that the closest point from the vector
811:
4121:
4125:
1850:
Therefore, the projection matrix (and hat matrix) is given by
351:
can also be expressed compactly using the projection matrix:
227:{\displaystyle \mathbf {\hat {y}} =\mathbf {P} \mathbf {y} .}
1621:
When the weights for each observation are identical and the
3446:. Then the projection matrix can be decomposed as follows:
726:{\displaystyle \mathbf {\Sigma } =\sigma ^{2}\mathbf {I} }
3800:, which might be too large to fit into computer memory.
3439:{\displaystyle \mathbf {M} :=\mathbf {I} -\mathbf {P} }
2544:{\displaystyle \mathbf {M} :=\mathbf {I} -\mathbf {P} }
1574:
is a vector of unknown parameters to be estimated, and
514:{\displaystyle \mathbf {M} :=\mathbf {I} -\mathbf {P} }
3697:{\displaystyle \mathbf {M} =\mathbf {I} -\mathbf {P} }
3259:
3784:
3762:
3736:
3710:
3650:
3556:
3455:
3392:
3298:
3245:
3223:
3174:
3152:
3096:
3058:
2970:
2915:
2879:
2857:
2835:
2797:
2761:
2742:{\displaystyle \operatorname {rank} (\mathbf {P} )=r}
2715:
2702:{\displaystyle \operatorname {rank} (\mathbf {X} )=r}
2675:
2643:
2618:
2581:
2559:
2519:
2497:
2442:
2390:
2313:
2291:
2226:
2110:
1974:
1859:
1733:
1634:
1544:
1495:
1409:
1387:
1365:
1343:
1321:
1296:
1073:
1007:
982:
957:
935:
913:
888:
866:
844:
822:
742:
699:
669:
571:
542:
489:
463:
360:
335:
302:
272:
266:
is usually pronounced "y-hat", the projection matrix
243:
195:
163:
141:
85:
50:
4993:
4917:
4863:
4799:
4633:
4551:
4497:
4436:
4160:
3906:Hoaglin, David C.; Welsch, Roy E. (February 1978).
4042:
3988:
3855:Applied Matrix Algebra in the Statistical Sciences
3792:
3770:
3744:
3718:
3696:
3636:
3539:
3438:
3378:
3284:
3231:
3190:, the projection matrix can be used to define the
3182:
3160:
3119:
3066:
3043:
2954:
2901:
2865:
2843:
2805:
2769:
2741:
2701:
2651:
2626:
2604:
2567:
2543:
2505:
2482:
2428:
2369:
2299:
2257:
2209:
2089:
1931:
1839:
1713:
1552:
1527:
1470:
1395:
1373:
1351:
1329:
1307:
1278:
1052:
990:
968:
943:
921:
896:
874:
852:
830:
795:
725:
677:
651:
550:
513:
471:
446:
343:
310:
280:
258:
226:
178:
149:
99:
64:
383:
250:
202:
170:
4049:. Cambridge: Harvard University Press. pp.
1625:are uncorrelated, the estimated parameters are
3129:locally weighted scatterplot smoothing (LOESS)
4137:
3901:
3899:
3529:
3501:
3386:. Similarly, define the residual operator as
3120:{\displaystyle \mathbf {P} ^{2}=\mathbf {P} }
2605:{\displaystyle \mathbf {P} ^{2}=\mathbf {P} }
111:(dependent variable values) to the vector of
8:
3960:
3958:
3956:
3292:. Define the hat or projection operator as
3127:. However, this is not always the case; in
2902:{\displaystyle \mathbf {PX} =\mathbf {X} ,}
4711:Fundamental (linear differential equation)
4144:
4130:
4122:
4100:(3rd ed.). Berlin: Springer. p.
3936:
3785:
3783:
3763:
3761:
3737:
3735:
3711:
3709:
3686:
3678:
3670:
3659:
3651:
3649:
3628:
3627:
3626:
3621:
3611:
3601:
3595:
3594:
3593:
3588:
3576:
3565:
3557:
3555:
3528:
3527:
3522:
3514:
3506:
3500:
3499:
3494:
3483:
3475:
3464:
3456:
3454:
3428:
3420:
3412:
3401:
3393:
3391:
3370:
3369:
3368:
3363:
3353:
3343:
3337:
3336:
3335:
3330:
3318:
3307:
3299:
3297:
3269:
3262:
3254:
3246:
3244:
3224:
3222:
3175:
3173:
3153:
3151:
3142:of the projection matrix is equal to the
3112:
3103:
3098:
3095:
3078:The projection matrix corresponding to a
3059:
3057:
3033:
3020:
3012:
3002:
2994:
2984:
2976:
2969:
2947:
2939:
2929:
2921:
2914:
2891:
2880:
2878:
2858:
2856:
2836:
2834:
2798:
2796:
2762:
2760:
2725:
2714:
2685:
2674:
2644:
2642:
2619:
2617:
2597:
2588:
2583:
2580:
2560:
2558:
2536:
2528:
2520:
2518:
2498:
2496:
2472:
2464:
2459:
2451:
2443:
2441:
2418:
2410:
2402:
2391:
2389:
2361:
2360:
2359:
2354:
2344:
2334:
2328:
2327:
2326:
2321:
2312:
2292:
2290:
2231:
2225:
2198:
2193:
2186:
2185:
2184:
2179:
2169:
2159:
2150:
2145:
2138:
2137:
2136:
2131:
2119:
2111:
2109:
2082:
2073:
2068:
2061:
2060:
2059:
2054:
2044:
2034:
2025:
2020:
2013:
2012:
2011:
2006:
1990:
1979:
1977:
1976:
1973:
1920:
1919:
1918:
1913:
1903:
1893:
1887:
1886:
1885:
1880:
1868:
1860:
1858:
1829:
1823:
1822:
1821:
1816:
1806:
1796:
1790:
1789:
1788:
1783:
1771:
1757:
1756:
1751:
1737:
1735:
1734:
1732:
1703:
1697:
1696:
1695:
1690:
1680:
1670:
1664:
1663:
1662:
1657:
1636:
1635:
1633:
1545:
1543:
1517:
1509:
1504:
1496:
1494:
1462:
1461:
1460:
1455:
1445:
1435:
1429:
1428:
1427:
1422:
1410:
1408:
1388:
1386:
1366:
1364:
1344:
1342:
1322:
1320:
1297:
1295:
1267:
1261:
1260:
1259:
1254:
1244:
1234:
1228:
1227:
1226:
1221:
1202:
1184:
1178:
1177:
1176:
1171:
1158:
1152:
1151:
1150:
1145:
1120:
1114:
1113:
1112:
1107:
1094:
1088:
1087:
1086:
1081:
1074:
1072:
1033:
1025:
1016:
1015:
1014:
1009:
1006:
983:
981:
958:
956:
936:
934:
914:
912:
889:
887:
867:
865:
845:
843:
823:
821:
787:
773:
765:
750:
749:
744:
741:
718:
712:
700:
698:
670:
668:
639:
631:
621:
615:
614:
613:
603:
595:
579:
578:
573:
570:
543:
541:
506:
498:
490:
488:
464:
462:
436:
426:
418:
405:
400:
392:
378:
377:
369:
361:
359:
336:
334:
303:
301:
273:
271:
245:
244:
242:
216:
211:
197:
196:
194:
165:
164:
162:
142:
140:
89:
84:
54:
49:
3908:"The Hat Matrix in Regression and ANOVA"
2265:, though now it is no longer symmetric.
115:(or predicted values). It describes the
5016:Matrix representation of conic sections
3970:Statistical Models: Theory and Practice
3844:
3205:, which are concerned with identifying
1759:
1638:
1518:
1510:
691:independent and identically distributed
3778:without explicitly forming the matrix
1943:Weighted and generalized least squares
3991:Data Fitting in the Chemical Sciences
7:
1337:, the projection matrix, which maps
259:{\displaystyle \mathbf {\hat {y}} }
179:{\displaystyle \mathbf {\hat {y}} }
157:and the vector of fitted values by
30:For the linear transformation, see
678:{\displaystyle \mathbf {\Sigma } }
25:
4096:Linear Models and Generalizations
4016:Draper, N. R.; Smith, H. (1998).
5050:
3786:
3764:
3738:
3712:
3687:
3679:
3671:
3660:
3652:
3622:
3602:
3589:
3577:
3566:
3558:
3523:
3515:
3507:
3495:
3484:
3476:
3465:
3457:
3429:
3421:
3413:
3402:
3394:
3364:
3344:
3331:
3319:
3308:
3300:
3270:
3263:
3247:
3239:can be decomposed by columns as
3225:
3176:
3154:
3113:
3099:
3074:is unique for certain subspaces.
3060:
3034:
3021:
3013:
3003:
2995:
2985:
2977:
2948:
2940:
2930:
2922:
2892:
2884:
2881:
2859:
2837:
2799:
2791:zeros, while the eigenvalues of
2763:
2726:
2686:
2645:
2620:
2598:
2584:
2561:
2537:
2529:
2521:
2499:
2473:
2465:
2460:
2452:
2444:
2419:
2411:
2403:
2392:
2355:
2335:
2322:
2293:
2258:{\displaystyle H^{2}=H\cdot H=H}
2194:
2180:
2160:
2146:
2132:
2120:
2112:
2083:
2069:
2055:
2035:
2021:
2007:
1914:
1894:
1881:
1869:
1861:
1830:
1817:
1797:
1784:
1772:
1752:
1738:
1704:
1691:
1671:
1658:
1546:
1505:
1497:
1456:
1436:
1423:
1411:
1389:
1367:
1345:
1323:
1301:
1298:
1268:
1255:
1235:
1222:
1203:
1188:
1185:
1172:
1159:
1146:
1124:
1121:
1108:
1095:
1082:
1064:From there, one rearranges, so
1037:
1034:
1026:
1010:
984:
962:
959:
937:
915:
890:
868:
846:
824:
774:
766:
751:
745:
719:
701:
671:
640:
632:
622:
604:
596:
580:
574:
544:
521:is sometimes referred to as the
507:
499:
491:
465:
437:
427:
419:
406:
401:
393:
380:
370:
362:
337:
304:
274:
247:
217:
212:
199:
167:
143:
90:
55:
4918:Used in science and engineering
2277:, the projection matrix is the
4161:Explicitly constrained entries
3852:Basilevsky, Alexander (2005).
3691:
3683:
3664:
3656:
3570:
3562:
3519:
3511:
3488:
3480:
3469:
3461:
3433:
3425:
3406:
3398:
3312:
3304:
2730:
2722:
2690:
2682:
2415:
2399:
2220:and again it may be seen that
1984:
1762:
1742:
1641:
1196:
1138:
1041:
1022:
326:The formula for the vector of
100:{\displaystyle (\mathbf {H} )}
94:
86:
65:{\displaystyle (\mathbf {P} )}
59:
51:
1:
4935:Fundamental (computer vision)
1308:{\displaystyle \mathbf {Ax} }
969:{\displaystyle \mathbf {Ax} }
3828:Effective degrees of freedom
3793:{\displaystyle \mathbf {X} }
3771:{\displaystyle \mathbf {X} }
3745:{\displaystyle \mathbf {A} }
3719:{\displaystyle \mathbf {A} }
3232:{\displaystyle \mathbf {X} }
3192:effective degrees of freedom
3183:{\displaystyle \mathbf {y} }
3161:{\displaystyle \mathbf {X} }
3067:{\displaystyle \mathbf {P} }
2866:{\displaystyle \mathbf {P} }
2844:{\displaystyle \mathbf {X} }
2806:{\displaystyle \mathbf {M} }
2770:{\displaystyle \mathbf {P} }
2652:{\displaystyle \mathbf {X} }
2627:{\displaystyle \mathbf {M} }
2568:{\displaystyle \mathbf {P} }
2506:{\displaystyle \mathbf {P} }
2300:{\displaystyle \mathbf {X} }
1553:{\displaystyle \mathbf {X} }
1396:{\displaystyle \mathbf {A} }
1374:{\displaystyle \mathbf {x} }
1352:{\displaystyle \mathbf {b} }
1330:{\displaystyle \mathbf {A} }
991:{\displaystyle \mathbf {A} }
944:{\displaystyle \mathbf {A} }
922:{\displaystyle \mathbf {b} }
897:{\displaystyle \mathbf {x} }
875:{\displaystyle \mathbf {A} }
853:{\displaystyle \mathbf {b} }
831:{\displaystyle \mathbf {A} }
551:{\displaystyle \mathbf {r} }
472:{\displaystyle \mathbf {I} }
344:{\displaystyle \mathbf {r} }
311:{\displaystyle \mathbf {y} }
281:{\displaystyle \mathbf {P} }
150:{\displaystyle \mathbf {y} }
72:, sometimes also called the
4701:Duplication and elimination
4500:eigenvalues or eigenvectors
4018:Applied Regression Analysis
3858:. Dover. pp. 160–176.
3833:Mean and predicted response
3818:Projection (linear algebra)
32:Projection (linear algebra)
5110:
4634:With specific applications
4263:Discrete Fourier Transform
3974:Cambridge University Press
3217:Suppose the design matrix
1946:
1614:
1315:is on the column space of
29:
5044:
4925:Cabibbo–Kobayashi–Maskawa
4552:Satisfying conditions on
4041:Amemiya, Takeshi (1985).
3916:The American Statistician
1953:Generalized least squares
1724:so the fitted values are
929:onto the column space of
860:onto the column space of
322:Application for residuals
3207:influential observations
2513:is symmetric, and so is
4283:Generalized permutation
2101:the hat matrix is thus
5057:Mathematics portal
3794:
3772:
3746:
3720:
3698:
3638:
3541:
3440:
3380:
3286:
3233:
3184:
3162:
3121:
3068:
3045:
2956:
2903:
2867:
2845:
2807:
2771:
2743:
2703:
2653:
2628:
2606:
2569:
2545:
2507:
2484:
2430:
2371:
2301:
2259:
2211:
2091:
1949:Weighted least squares
1933:
1841:
1715:
1617:Ordinary least squares
1611:Ordinary least squares
1554:
1529:
1472:
1397:
1375:
1353:
1331:
1309:
1280:
1054:
992:
970:
945:
923:
904:
898:
876:
854:
832:
797:
727:
679:
653:
552:
515:
473:
448:
345:
312:
282:
260:
228:
180:
151:
101:
66:
4045:Advanced Econometrics
3823:Studentized residuals
3795:
3773:
3747:
3721:
3699:
3639:
3542:
3441:
3381:
3287:
3234:
3185:
3163:
3122:
3069:
3046:
2957:
2904:
2868:
2846:
2808:
2772:
2744:
2704:
2654:
2629:
2607:
2570:
2546:
2508:
2485:
2431:
2372:
2302:
2285:of the design matrix
2279:orthogonal projection
2260:
2212:
2092:
1947:Further information:
1934:
1842:
1716:
1615:Further information:
1580:is the error vector.
1562:explanatory variables
1555:
1530:
1473:
1398:
1376:
1354:
1332:
1310:
1281:
1055:
993:
971:
946:
924:
899:
877:
855:
833:
815:
798:
728:
680:
654:
553:
523:residual maker matrix
516:
474:
449:
346:
313:
283:
261:
229:
181:
152:
107:, maps the vector of
102:
67:
27:Concept in statistics
3782:
3760:
3734:
3708:
3648:
3554:
3453:
3390:
3296:
3243:
3221:
3172:
3150:
3094:
3056:
2968:
2913:
2877:
2855:
2833:
2795:
2759:
2713:
2673:
2641:
2616:
2579:
2557:
2517:
2495:
2440:
2388:
2311:
2289:
2224:
2108:
1972:
1857:
1731:
1632:
1585:linear least squares
1542:
1493:
1407:
1385:
1363:
1341:
1319:
1294:
1071:
1005:
980:
955:
933:
911:
886:
864:
842:
820:
740:
697:
667:
569:
540:
487:
461:
358:
333:
300:
270:
241:
193:
161:
139:
83:
48:
5089:Regression analysis
5006:Linear independence
4253:Diagonally dominant
3728:fixed effects model
2851:is invariant under
733:, this reduces to:
5011:Matrix exponential
5001:Jordan normal form
4835:Fisher information
4706:Euclidean distance
4620:Totally unimodular
3790:
3768:
3742:
3716:
3694:
3634:
3537:
3436:
3376:
3282:
3276:
3229:
3180:
3158:
3117:
3064:
3041:
2952:
2899:
2863:
2841:
2803:
2767:
2739:
2699:
2649:
2624:
2602:
2565:
2541:
2503:
2480:
2426:
2379:pseudoinverse of X
2367:
2297:
2255:
2207:
2087:
1929:
1837:
1711:
1593:regression splines
1550:
1525:
1468:
1393:
1371:
1349:
1327:
1305:
1276:
1274:
1050:
988:
966:
941:
919:
905:
894:
872:
850:
828:
793:
723:
675:
649:
548:
527:annihilator matrix
511:
469:
444:
341:
308:
278:
256:
224:
176:
147:
97:
62:
5076:
5075:
5068:Category:Matrices
4940:Fuzzy associative
4830:Doubly stochastic
4538:Positive-definite
4218:Block tridiagonal
4111:978-3-540-74226-5
4081:. April 13, 2017.
3987:Gans, P. (1992).
3966:David A. Freedman
3630:
3597:
3372:
3339:
3213:Blockwise formula
2363:
2330:
2188:
2140:
2063:
2015:
1993:
1987:
1961:of the errors is
1959:covariance matrix
1922:
1889:
1825:
1792:
1765:
1745:
1699:
1666:
1644:
1601:kernel regression
1589:smoothing splines
1464:
1431:
1290:Therefore, since
1263:
1230:
1180:
1154:
1116:
1090:
1018:
687:covariance matrix
617:
560:error propagation
536:of the residuals
534:covariance matrix
386:
253:
205:
173:
131:If the vector of
43:projection matrix
16:(Redirected from
5101:
5063:List of matrices
5055:
5054:
5031:Row echelon form
4975:State transition
4904:Seidel adjacency
4786:Totally positive
4646:Alternating sign
4243:Complex Hadamard
4146:
4139:
4132:
4123:
4116:
4115:
4099:
4089:
4083:
4082:
4071:
4065:
4064:
4048:
4038:
4032:
4031:
4013:
4007:
4006:
3994:
3984:
3978:
3977:
3962:
3951:
3950:
3940:
3912:
3903:
3894:
3893:
3891:
3885:. Archived from
3884:
3876:
3870:
3869:
3849:
3799:
3797:
3796:
3791:
3789:
3777:
3775:
3774:
3769:
3767:
3751:
3749:
3748:
3743:
3741:
3725:
3723:
3722:
3717:
3715:
3703:
3701:
3700:
3695:
3690:
3682:
3674:
3663:
3655:
3643:
3641:
3640:
3635:
3633:
3632:
3631:
3625:
3619:
3618:
3610:
3606:
3605:
3600:
3599:
3598:
3592:
3580:
3569:
3561:
3546:
3544:
3543:
3538:
3533:
3532:
3526:
3518:
3510:
3505:
3504:
3498:
3487:
3479:
3468:
3460:
3445:
3443:
3442:
3437:
3432:
3424:
3416:
3405:
3397:
3385:
3383:
3382:
3377:
3375:
3374:
3373:
3367:
3361:
3360:
3352:
3348:
3347:
3342:
3341:
3340:
3334:
3322:
3311:
3303:
3291:
3289:
3288:
3283:
3281:
3280:
3273:
3266:
3250:
3238:
3236:
3235:
3230:
3228:
3189:
3187:
3186:
3181:
3179:
3167:
3165:
3164:
3159:
3157:
3126:
3124:
3123:
3118:
3116:
3108:
3107:
3102:
3073:
3071:
3070:
3065:
3063:
3050:
3048:
3047:
3042:
3037:
3029:
3025:
3024:
3016:
3006:
2998:
2993:
2989:
2988:
2980:
2961:
2959:
2958:
2953:
2951:
2943:
2938:
2934:
2933:
2925:
2908:
2906:
2905:
2900:
2895:
2887:
2872:
2870:
2869:
2864:
2862:
2850:
2848:
2847:
2842:
2840:
2822:
2812:
2810:
2809:
2804:
2802:
2790:
2776:
2774:
2773:
2768:
2766:
2748:
2746:
2745:
2740:
2729:
2708:
2706:
2705:
2700:
2689:
2668:
2658:
2656:
2655:
2650:
2648:
2633:
2631:
2630:
2625:
2623:
2611:
2609:
2608:
2603:
2601:
2593:
2592:
2587:
2574:
2572:
2571:
2566:
2564:
2550:
2548:
2547:
2542:
2540:
2532:
2524:
2512:
2510:
2509:
2504:
2502:
2489:
2487:
2486:
2481:
2476:
2468:
2463:
2455:
2447:
2435:
2433:
2432:
2427:
2422:
2414:
2406:
2395:
2376:
2374:
2373:
2368:
2366:
2365:
2364:
2358:
2352:
2351:
2343:
2339:
2338:
2333:
2332:
2331:
2325:
2306:
2304:
2303:
2298:
2296:
2264:
2262:
2261:
2256:
2236:
2235:
2216:
2214:
2213:
2208:
2206:
2205:
2197:
2191:
2190:
2189:
2183:
2177:
2176:
2168:
2164:
2163:
2158:
2157:
2149:
2143:
2142:
2141:
2135:
2123:
2115:
2096:
2094:
2093:
2088:
2086:
2081:
2080:
2072:
2066:
2065:
2064:
2058:
2052:
2051:
2043:
2039:
2038:
2033:
2032:
2024:
2018:
2017:
2016:
2010:
1995:
1994:
1991:
1989:
1988:
1983:
1978:
1938:
1936:
1935:
1930:
1925:
1924:
1923:
1917:
1911:
1910:
1902:
1898:
1897:
1892:
1891:
1890:
1884:
1872:
1864:
1846:
1844:
1843:
1838:
1833:
1828:
1827:
1826:
1820:
1814:
1813:
1805:
1801:
1800:
1795:
1794:
1793:
1787:
1775:
1767:
1766:
1758:
1755:
1747:
1746:
1741:
1736:
1720:
1718:
1717:
1712:
1707:
1702:
1701:
1700:
1694:
1688:
1687:
1679:
1675:
1674:
1669:
1668:
1667:
1661:
1646:
1645:
1637:
1605:linear filtering
1597:local regression
1559:
1557:
1556:
1551:
1549:
1534:
1532:
1531:
1526:
1521:
1513:
1508:
1500:
1477:
1475:
1474:
1469:
1467:
1466:
1465:
1459:
1453:
1452:
1444:
1440:
1439:
1434:
1433:
1432:
1426:
1414:
1402:
1400:
1399:
1394:
1392:
1380:
1378:
1377:
1372:
1370:
1358:
1356:
1355:
1350:
1348:
1336:
1334:
1333:
1328:
1326:
1314:
1312:
1311:
1306:
1304:
1285:
1283:
1282:
1277:
1275:
1271:
1266:
1265:
1264:
1258:
1252:
1251:
1243:
1239:
1238:
1233:
1232:
1231:
1225:
1206:
1200:
1191:
1183:
1182:
1181:
1175:
1162:
1157:
1156:
1155:
1149:
1142:
1127:
1119:
1118:
1117:
1111:
1098:
1093:
1092:
1091:
1085:
1078:
1077:
1059:
1057:
1056:
1051:
1040:
1029:
1021:
1020:
1019:
1013:
997:
995:
994:
989:
987:
975:
973:
972:
967:
965:
950:
948:
947:
942:
940:
928:
926:
925:
920:
918:
903:
901:
900:
895:
893:
881:
879:
878:
873:
871:
859:
857:
856:
851:
849:
837:
835:
834:
829:
827:
802:
800:
799:
794:
792:
791:
782:
778:
777:
769:
756:
755:
754:
748:
732:
730:
729:
724:
722:
717:
716:
704:
693:errors in which
684:
682:
681:
676:
674:
658:
656:
655:
650:
648:
644:
643:
635:
625:
620:
619:
618:
612:
608:
607:
599:
585:
584:
583:
577:
557:
555:
554:
549:
547:
520:
518:
517:
512:
510:
502:
494:
478:
476:
475:
470:
468:
453:
451:
450:
445:
440:
435:
431:
430:
422:
409:
404:
396:
388:
387:
379:
373:
365:
350:
348:
347:
342:
340:
317:
315:
314:
309:
307:
287:
285:
284:
279:
277:
265:
263:
262:
257:
255:
254:
246:
233:
231:
230:
225:
220:
215:
207:
206:
198:
185:
183:
182:
177:
175:
174:
166:
156:
154:
153:
148:
146:
106:
104:
103:
98:
93:
74:influence matrix
71:
69:
68:
63:
58:
21:
5109:
5108:
5104:
5103:
5102:
5100:
5099:
5098:
5079:
5078:
5077:
5072:
5049:
5040:
4989:
4913:
4859:
4795:
4629:
4547:
4493:
4432:
4233:Centrosymmetric
4156:
4150:
4120:
4119:
4112:
4091:
4090:
4086:
4073:
4072:
4068:
4061:
4040:
4039:
4035:
4028:
4015:
4014:
4010:
4003:
3986:
3985:
3981:
3964:
3963:
3954:
3929:10.2307/2683469
3910:
3905:
3904:
3897:
3889:
3882:
3878:
3877:
3873:
3866:
3851:
3850:
3846:
3841:
3814:
3806:
3780:
3779:
3758:
3757:
3732:
3731:
3706:
3705:
3646:
3645:
3620:
3587:
3586:
3582:
3581:
3552:
3551:
3451:
3450:
3388:
3387:
3362:
3329:
3328:
3324:
3323:
3294:
3293:
3275:
3274:
3267:
3255:
3241:
3240:
3219:
3218:
3215:
3203:Cook's distance
3170:
3169:
3148:
3147:
3097:
3092:
3091:
3054:
3053:
3011:
3007:
2975:
2971:
2966:
2965:
2920:
2916:
2911:
2910:
2875:
2874:
2853:
2852:
2831:
2830:
2814:
2793:
2792:
2782:
2757:
2756:
2711:
2710:
2671:
2670:
2660:
2639:
2638:
2614:
2613:
2582:
2577:
2576:
2575:is idempotent:
2555:
2554:
2515:
2514:
2493:
2492:
2438:
2437:
2386:
2385:
2353:
2320:
2319:
2315:
2314:
2309:
2308:
2287:
2286:
2271:
2227:
2222:
2221:
2192:
2178:
2144:
2130:
2129:
2125:
2124:
2106:
2105:
2067:
2053:
2019:
2005:
2004:
2000:
1999:
1975:
1970:
1969:
1955:
1945:
1912:
1879:
1878:
1874:
1873:
1855:
1854:
1815:
1782:
1781:
1777:
1776:
1729:
1728:
1689:
1656:
1655:
1651:
1650:
1630:
1629:
1619:
1613:
1560:is a matrix of
1540:
1539:
1491:
1490:
1484:
1454:
1421:
1420:
1416:
1415:
1405:
1404:
1383:
1382:
1361:
1360:
1339:
1338:
1317:
1316:
1292:
1291:
1273:
1272:
1253:
1220:
1219:
1215:
1214:
1207:
1199:
1193:
1192:
1170:
1163:
1144:
1141:
1135:
1134:
1106:
1099:
1080:
1069:
1068:
1008:
1003:
1002:
978:
977:
953:
952:
931:
930:
909:
908:
884:
883:
862:
861:
840:
839:
818:
817:
810:
783:
764:
760:
743:
738:
737:
708:
695:
694:
665:
664:
630:
626:
594:
590:
589:
572:
567:
566:
538:
537:
485:
484:
481:identity matrix
459:
458:
417:
413:
356:
355:
331:
330:
324:
298:
297:
268:
267:
239:
238:
191:
190:
159:
158:
137:
136:
133:response values
129:
109:response values
81:
80:
46:
45:
35:
28:
23:
22:
18:Operator matrix
15:
12:
11:
5:
5107:
5105:
5097:
5096:
5091:
5081:
5080:
5074:
5073:
5071:
5070:
5065:
5060:
5045:
5042:
5041:
5039:
5038:
5033:
5028:
5023:
5021:Perfect matrix
5018:
5013:
5008:
5003:
4997:
4995:
4991:
4990:
4988:
4987:
4982:
4977:
4972:
4967:
4962:
4957:
4952:
4947:
4942:
4937:
4932:
4927:
4921:
4919:
4915:
4914:
4912:
4911:
4906:
4901:
4896:
4891:
4886:
4881:
4876:
4870:
4868:
4861:
4860:
4858:
4857:
4852:
4847:
4842:
4837:
4832:
4827:
4822:
4817:
4812:
4806:
4804:
4797:
4796:
4794:
4793:
4791:Transformation
4788:
4783:
4778:
4773:
4768:
4763:
4758:
4753:
4748:
4743:
4738:
4733:
4728:
4723:
4718:
4713:
4708:
4703:
4698:
4693:
4688:
4683:
4678:
4673:
4668:
4663:
4658:
4653:
4648:
4643:
4637:
4635:
4631:
4630:
4628:
4627:
4622:
4617:
4612:
4607:
4602:
4597:
4592:
4587:
4582:
4577:
4568:
4562:
4560:
4549:
4548:
4546:
4545:
4540:
4535:
4530:
4528:Diagonalizable
4525:
4520:
4515:
4510:
4504:
4502:
4498:Conditions on
4495:
4494:
4492:
4491:
4486:
4481:
4476:
4471:
4466:
4461:
4456:
4451:
4446:
4440:
4438:
4434:
4433:
4431:
4430:
4425:
4420:
4415:
4410:
4405:
4400:
4395:
4390:
4385:
4380:
4378:Skew-symmetric
4375:
4373:Skew-Hermitian
4370:
4365:
4360:
4355:
4350:
4345:
4340:
4335:
4330:
4325:
4320:
4315:
4310:
4305:
4300:
4295:
4290:
4285:
4280:
4275:
4270:
4265:
4260:
4255:
4250:
4245:
4240:
4235:
4230:
4225:
4220:
4215:
4210:
4208:Block-diagonal
4205:
4200:
4195:
4190:
4185:
4183:Anti-symmetric
4180:
4178:Anti-Hermitian
4175:
4170:
4164:
4162:
4158:
4157:
4151:
4149:
4148:
4141:
4134:
4126:
4118:
4117:
4110:
4084:
4079:Stack Exchange
4066:
4059:
4033:
4026:
4008:
4001:
3979:
3952:
3895:
3892:on 2014-09-03.
3871:
3864:
3843:
3842:
3840:
3837:
3836:
3835:
3830:
3825:
3820:
3813:
3810:
3805:
3802:
3788:
3766:
3740:
3714:
3693:
3689:
3685:
3681:
3677:
3673:
3669:
3666:
3662:
3658:
3654:
3624:
3617:
3614:
3609:
3604:
3591:
3585:
3579:
3575:
3572:
3568:
3564:
3560:
3548:
3547:
3536:
3531:
3525:
3521:
3517:
3513:
3509:
3503:
3497:
3493:
3490:
3486:
3482:
3478:
3474:
3471:
3467:
3463:
3459:
3435:
3431:
3427:
3423:
3419:
3415:
3411:
3408:
3404:
3400:
3396:
3366:
3359:
3356:
3351:
3346:
3333:
3327:
3321:
3317:
3314:
3310:
3306:
3302:
3279:
3272:
3268:
3265:
3261:
3260:
3258:
3253:
3249:
3227:
3214:
3211:
3194:of the model.
3178:
3156:
3115:
3111:
3106:
3101:
3076:
3075:
3062:
3051:
3040:
3036:
3032:
3028:
3023:
3019:
3015:
3010:
3005:
3001:
2997:
2992:
2987:
2983:
2979:
2974:
2963:
2950:
2946:
2942:
2937:
2932:
2928:
2924:
2919:
2898:
2894:
2890:
2886:
2883:
2861:
2839:
2828:
2801:
2765:
2749:
2738:
2735:
2732:
2728:
2724:
2721:
2718:
2698:
2695:
2692:
2688:
2684:
2681:
2678:
2647:
2635:
2622:
2600:
2596:
2591:
2586:
2563:
2552:
2539:
2535:
2531:
2527:
2523:
2501:
2490:
2479:
2475:
2471:
2467:
2462:
2458:
2454:
2450:
2446:
2425:
2421:
2417:
2413:
2409:
2405:
2401:
2398:
2394:
2357:
2350:
2347:
2342:
2337:
2324:
2318:
2295:
2275:linear algebra
2270:
2267:
2254:
2251:
2248:
2245:
2242:
2239:
2234:
2230:
2218:
2217:
2204:
2201:
2196:
2182:
2175:
2172:
2167:
2162:
2156:
2153:
2148:
2134:
2128:
2122:
2118:
2114:
2099:
2098:
2085:
2079:
2076:
2071:
2057:
2050:
2047:
2042:
2037:
2031:
2028:
2023:
2009:
2003:
1998:
1986:
1982:
1965:. Then since
1944:
1941:
1940:
1939:
1928:
1916:
1909:
1906:
1901:
1896:
1883:
1877:
1871:
1867:
1863:
1848:
1847:
1836:
1832:
1819:
1812:
1809:
1804:
1799:
1786:
1780:
1774:
1770:
1764:
1761:
1754:
1750:
1744:
1740:
1722:
1721:
1710:
1706:
1693:
1686:
1683:
1678:
1673:
1660:
1654:
1649:
1643:
1640:
1612:
1609:
1548:
1536:
1535:
1524:
1520:
1516:
1512:
1507:
1503:
1499:
1483:
1480:
1458:
1451:
1448:
1443:
1438:
1425:
1419:
1413:
1391:
1369:
1347:
1325:
1303:
1300:
1288:
1287:
1270:
1257:
1250:
1247:
1242:
1237:
1224:
1218:
1213:
1210:
1208:
1205:
1201:
1198:
1195:
1194:
1190:
1187:
1174:
1169:
1166:
1164:
1161:
1148:
1143:
1140:
1137:
1136:
1133:
1130:
1126:
1123:
1110:
1105:
1102:
1100:
1097:
1084:
1079:
1076:
1062:
1061:
1049:
1046:
1043:
1039:
1036:
1032:
1028:
1024:
1012:
986:
964:
961:
939:
917:
892:
882:is the vector
870:
848:
826:
809:
806:
805:
804:
790:
786:
781:
776:
772:
768:
763:
759:
753:
747:
721:
715:
711:
707:
703:
673:
661:
660:
647:
642:
638:
634:
629:
624:
611:
606:
602:
598:
593:
588:
582:
576:
546:
509:
505:
501:
497:
493:
467:
455:
454:
443:
439:
434:
429:
425:
421:
416:
412:
408:
403:
399:
395:
391:
385:
382:
376:
372:
368:
364:
339:
323:
320:
306:
292:as it "puts a
288:is also named
276:
252:
249:
235:
234:
223:
219:
214:
210:
204:
201:
172:
169:
145:
135:is denoted by
128:
125:
96:
92:
88:
61:
57:
53:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5106:
5095:
5092:
5090:
5087:
5086:
5084:
5069:
5066:
5064:
5061:
5059:
5058:
5053:
5047:
5046:
5043:
5037:
5034:
5032:
5029:
5027:
5026:Pseudoinverse
5024:
5022:
5019:
5017:
5014:
5012:
5009:
5007:
5004:
5002:
4999:
4998:
4996:
4994:Related terms
4992:
4986:
4985:Z (chemistry)
4983:
4981:
4978:
4976:
4973:
4971:
4968:
4966:
4963:
4961:
4958:
4956:
4953:
4951:
4948:
4946:
4943:
4941:
4938:
4936:
4933:
4931:
4928:
4926:
4923:
4922:
4920:
4916:
4910:
4907:
4905:
4902:
4900:
4897:
4895:
4892:
4890:
4887:
4885:
4882:
4880:
4877:
4875:
4872:
4871:
4869:
4867:
4862:
4856:
4853:
4851:
4848:
4846:
4843:
4841:
4838:
4836:
4833:
4831:
4828:
4826:
4823:
4821:
4818:
4816:
4813:
4811:
4808:
4807:
4805:
4803:
4798:
4792:
4789:
4787:
4784:
4782:
4779:
4777:
4774:
4772:
4769:
4767:
4764:
4762:
4759:
4757:
4754:
4752:
4749:
4747:
4744:
4742:
4739:
4737:
4734:
4732:
4729:
4727:
4724:
4722:
4719:
4717:
4714:
4712:
4709:
4707:
4704:
4702:
4699:
4697:
4694:
4692:
4689:
4687:
4684:
4682:
4679:
4677:
4674:
4672:
4669:
4667:
4664:
4662:
4659:
4657:
4654:
4652:
4649:
4647:
4644:
4642:
4639:
4638:
4636:
4632:
4626:
4623:
4621:
4618:
4616:
4613:
4611:
4608:
4606:
4603:
4601:
4598:
4596:
4593:
4591:
4588:
4586:
4583:
4581:
4578:
4576:
4572:
4569:
4567:
4564:
4563:
4561:
4559:
4555:
4550:
4544:
4541:
4539:
4536:
4534:
4531:
4529:
4526:
4524:
4521:
4519:
4516:
4514:
4511:
4509:
4506:
4505:
4503:
4501:
4496:
4490:
4487:
4485:
4482:
4480:
4477:
4475:
4472:
4470:
4467:
4465:
4462:
4460:
4457:
4455:
4452:
4450:
4447:
4445:
4442:
4441:
4439:
4435:
4429:
4426:
4424:
4421:
4419:
4416:
4414:
4411:
4409:
4406:
4404:
4401:
4399:
4396:
4394:
4391:
4389:
4386:
4384:
4381:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4359:
4356:
4354:
4351:
4349:
4346:
4344:
4343:Pentadiagonal
4341:
4339:
4336:
4334:
4331:
4329:
4326:
4324:
4321:
4319:
4316:
4314:
4311:
4309:
4306:
4304:
4301:
4299:
4296:
4294:
4291:
4289:
4286:
4284:
4281:
4279:
4276:
4274:
4271:
4269:
4266:
4264:
4261:
4259:
4256:
4254:
4251:
4249:
4246:
4244:
4241:
4239:
4236:
4234:
4231:
4229:
4226:
4224:
4221:
4219:
4216:
4214:
4211:
4209:
4206:
4204:
4201:
4199:
4196:
4194:
4191:
4189:
4186:
4184:
4181:
4179:
4176:
4174:
4173:Anti-diagonal
4171:
4169:
4166:
4165:
4163:
4159:
4154:
4147:
4142:
4140:
4135:
4133:
4128:
4127:
4124:
4113:
4107:
4103:
4098:
4097:
4088:
4085:
4080:
4076:
4070:
4067:
4062:
4060:0-674-00560-0
4056:
4052:
4047:
4046:
4037:
4034:
4029:
4027:0-471-17082-8
4023:
4019:
4012:
4009:
4004:
4002:0-471-93412-7
3998:
3993:
3992:
3983:
3980:
3975:
3971:
3967:
3961:
3959:
3957:
3953:
3948:
3944:
3939:
3934:
3930:
3926:
3922:
3918:
3917:
3909:
3902:
3900:
3896:
3888:
3881:
3875:
3872:
3867:
3865:0-486-44538-0
3861:
3857:
3856:
3848:
3845:
3838:
3834:
3831:
3829:
3826:
3824:
3821:
3819:
3816:
3815:
3811:
3809:
3803:
3801:
3755:
3754:sparse matrix
3729:
3675:
3667:
3615:
3612:
3607:
3583:
3573:
3550:where, e.g.,
3534:
3491:
3472:
3449:
3448:
3447:
3417:
3409:
3357:
3354:
3349:
3325:
3315:
3277:
3256:
3251:
3212:
3210:
3208:
3204:
3200:
3195:
3193:
3145:
3141:
3137:
3136:linear models
3132:
3130:
3109:
3104:
3089:
3085:
3081:
3052:
3038:
3030:
3026:
3017:
3008:
2999:
2990:
2981:
2972:
2964:
2944:
2935:
2926:
2917:
2896:
2888:
2829:
2826:
2821:
2817:
2789:
2785:
2780:
2754:
2750:
2736:
2733:
2719:
2716:
2696:
2693:
2679:
2676:
2667:
2663:
2636:
2594:
2589:
2553:
2533:
2525:
2491:
2477:
2469:
2456:
2448:
2423:
2407:
2396:
2384:
2383:
2382:
2380:
2348:
2345:
2340:
2316:
2307:. (Note that
2284:
2280:
2276:
2268:
2266:
2252:
2249:
2246:
2243:
2240:
2237:
2232:
2228:
2202:
2199:
2173:
2170:
2165:
2154:
2151:
2126:
2116:
2104:
2103:
2102:
2077:
2074:
2048:
2045:
2040:
2029:
2026:
2001:
1996:
1980:
1968:
1967:
1966:
1964:
1960:
1954:
1950:
1942:
1926:
1907:
1904:
1899:
1875:
1865:
1853:
1852:
1851:
1834:
1810:
1807:
1802:
1778:
1768:
1748:
1727:
1726:
1725:
1708:
1684:
1681:
1676:
1652:
1647:
1628:
1627:
1626:
1624:
1618:
1610:
1608:
1606:
1602:
1598:
1594:
1590:
1586:
1581:
1579:
1578:
1573:
1572:
1567:
1566:design matrix
1563:
1522:
1514:
1501:
1489:
1488:
1487:
1481:
1479:
1449:
1446:
1441:
1417:
1248:
1245:
1240:
1216:
1211:
1209:
1167:
1165:
1131:
1128:
1103:
1101:
1067:
1066:
1065:
1047:
1044:
1030:
1001:
1000:
999:
814:
807:
788:
784:
779:
770:
761:
757:
736:
735:
734:
713:
709:
705:
692:
688:
645:
636:
627:
609:
600:
591:
586:
565:
564:
563:
561:
535:
530:
528:
524:
503:
495:
483:. The matrix
482:
441:
432:
423:
414:
410:
397:
389:
374:
366:
354:
353:
352:
329:
321:
319:
295:
291:
221:
208:
189:
188:
187:
134:
126:
124:
122:
118:
114:
113:fitted values
110:
79:
75:
44:
40:
33:
19:
5048:
4980:Substitution
4866:graph theory
4839:
4363:Quaternionic
4353:Persymmetric
4095:
4087:
4078:
4069:
4044:
4036:
4017:
4011:
3990:
3982:
3969:
3923:(1): 17–22.
3920:
3914:
3887:the original
3874:
3854:
3847:
3807:
3549:
3216:
3196:
3133:
3080:linear model
3077:
2824:
2819:
2815:
2787:
2783:
2778:
2669:matrix with
2665:
2661:
2612:, and so is
2283:column space
2272:
2219:
2100:
1962:
1956:
1849:
1723:
1620:
1582:
1576:
1575:
1570:
1569:
1537:
1485:
1482:Linear model
1289:
1063:
906:
662:
531:
526:
522:
456:
325:
289:
236:
130:
77:
73:
42:
36:
4955:Hamiltonian
4879:Biadjacency
4815:Correlation
4731:Householder
4681:Commutation
4418:Vandermonde
4413:Tridiagonal
4348:Permutation
4338:Nonnegative
4323:Matrix unit
4203:Bisymmetric
3938:1721.1/1920
3752:is a large
3090:, that is,
2813:consist of
2777:consist of
2753:eigenvalues
5083:Categories
4855:Transition
4850:Stochastic
4820:Covariance
4802:statistics
4781:Symplectic
4776:Similarity
4605:Unimodular
4600:Orthogonal
4585:Involutory
4580:Invertible
4575:Projection
4571:Idempotent
4513:Convergent
4408:Triangular
4358:Polynomial
4303:Hessenberg
4273:Equivalent
4268:Elementary
4248:Copositive
4238:Conference
4198:Bidiagonal
3839:References
3088:idempotent
2269:Properties
816:A matrix,
290:hat matrix
127:Definition
78:hat matrix
39:statistics
5036:Wronskian
4960:Irregular
4950:Gell-Mann
4899:Laplacian
4894:Incidence
4874:Adjacency
4845:Precision
4810:Centering
4716:Generator
4686:Confusion
4671:Circulant
4651:Augmented
4610:Unipotent
4590:Nilpotent
4566:Congruent
4543:Stieltjes
4518:Defective
4508:Companion
4479:Redheffer
4398:Symmetric
4393:Sylvester
4368:Signature
4298:Hermitian
4278:Frobenius
4188:Arrowhead
4168:Alternant
4020:. Wiley.
3995:. Wiley.
3676:−
3613:−
3418:−
3355:−
3084:symmetric
3018:−
2982:−
2927:−
2823:ones and
2781:ones and
2720:
2680:
2534:−
2470:⊥
2457:−
2408:−
2346:−
2281:onto the
2244:⋅
2200:−
2195:Σ
2171:−
2152:−
2147:Σ
2075:−
2070:Σ
2046:−
2027:−
2022:Σ
1985:^
1981:β
1905:−
1808:−
1763:^
1760:β
1743:^
1682:−
1642:^
1639:β
1519:ε
1511:β
1447:−
1246:−
1197:⇒
1139:⇒
1104:−
1031:−
808:Intuition
785:σ
771:−
746:Σ
710:σ
702:Σ
672:Σ
637:−
623:Σ
601:−
575:Σ
562:, equals
504:−
424:−
398:−
384:^
375:−
328:residuals
251:^
203:^
171:^
121:leverages
117:influence
5094:Matrices
4864:Used in
4800:Used in
4761:Rotation
4736:Jacobian
4696:Distance
4676:Cofactor
4661:Carleman
4641:Adjugate
4625:Weighing
4558:inverses
4554:products
4523:Definite
4454:Identity
4444:Exchange
4437:Constant
4403:Toeplitz
4288:Hadamard
4258:Diagonal
3968:(2009).
3812:See also
3730:, where
3199:leverage
2873: :
1381:is just
4965:Overlap
4930:Density
4889:Edmonds
4766:Seifert
4726:Hessian
4691:Coxeter
4615:Unitary
4533:Hurwitz
4464:Of ones
4449:Hilbert
4383:Skyline
4328:Metzler
4318:Logical
4313:Integer
4223:Boolean
4155:classes
3947:2683469
3804:History
2709:, then
2377:is the
685:is the
525:or the
479:is the
4884:Degree
4825:Design
4756:Random
4746:Payoff
4741:Moment
4666:Cartan
4656:Bézout
4595:Normal
4469:Pascal
4459:Lehmer
4388:Sparse
4308:Hollow
4293:Hankel
4228:Cauchy
4153:Matrix
4108:
4057:
4053:–461.
4024:
3999:
3945:
3862:
3138:, the
2909:hence
2827:zeros.
2659:is an
1623:errors
1603:, and
1538:where
663:where
457:where
41:, the
4945:Gamma
4909:Tutte
4771:Shear
4484:Shift
4474:Pauli
4423:Walsh
4333:Moore
4213:Block
3943:JSTOR
3911:(PDF)
3890:(PDF)
3883:(PDF)
3140:trace
1564:(the
1403:, or
1359:onto
951:, is
558:, by
4751:Pick
4721:Gram
4489:Zero
4193:Band
4106:ISBN
4055:ISBN
4022:ISBN
3997:ISBN
3860:ISBN
3644:and
3201:and
3144:rank
3134:For
3086:and
2751:The
2717:rank
2677:rank
2436:and
1951:and
532:The
4840:Hat
4573:or
4556:or
4102:323
4051:460
3933:hdl
3925:doi
3146:of
3082:is
2755:of
2637:If
1992:GLS
1568:),
529:.
318:".
296:on
294:hat
237:As
76:or
37:In
5085::
4104:.
4077:.
3972:.
3955:^
3941:.
3931:.
3921:32
3919:.
3913:.
3898:^
3410::=
3316::=
2818:−
2786:−
2664:×
2526::=
1866::=
1607:.
1599:,
1595:,
1591:,
1587:,
1478:.
496::=
186:,
4970:S
4428:Z
4145:e
4138:t
4131:v
4114:.
4063:.
4030:.
4005:.
3976:.
3949:.
3935::
3927::
3868:.
3787:X
3765:X
3739:A
3713:A
3692:]
3688:A
3684:[
3680:P
3672:I
3668:=
3665:]
3661:A
3657:[
3653:M
3629:T
3623:A
3616:1
3608:)
3603:A
3596:T
3590:A
3584:(
3578:A
3574:=
3571:]
3567:A
3563:[
3559:P
3535:,
3530:]
3524:B
3520:]
3516:A
3512:[
3508:M
3502:[
3496:P
3492:+
3489:]
3485:A
3481:[
3477:P
3473:=
3470:]
3466:X
3462:[
3458:P
3434:]
3430:X
3426:[
3422:P
3414:I
3407:]
3403:X
3399:[
3395:M
3371:T
3365:X
3358:1
3350:)
3345:X
3338:T
3332:X
3326:(
3320:X
3313:]
3309:X
3305:[
3301:P
3278:]
3271:B
3264:A
3257:[
3252:=
3248:X
3226:X
3177:y
3155:X
3114:P
3110:=
3105:2
3100:P
3061:P
3039:.
3035:0
3031:=
3027:)
3022:P
3014:I
3009:(
3004:P
3000:=
2996:P
2991:)
2986:P
2978:I
2973:(
2962:.
2949:0
2945:=
2941:X
2936:)
2931:P
2923:I
2918:(
2897:,
2893:X
2889:=
2885:X
2882:P
2860:P
2838:X
2825:r
2820:r
2816:n
2800:M
2788:r
2784:n
2779:r
2764:P
2737:r
2734:=
2731:)
2727:P
2723:(
2697:r
2694:=
2691:)
2687:X
2683:(
2666:r
2662:n
2646:X
2634:.
2621:M
2599:P
2595:=
2590:2
2585:P
2562:P
2551:.
2538:P
2530:I
2522:M
2500:P
2478:.
2474:X
2466:y
2461:P
2453:y
2449:=
2445:u
2424:,
2420:y
2416:)
2412:P
2404:I
2400:(
2397:=
2393:u
2362:T
2356:X
2349:1
2341:)
2336:X
2329:T
2323:X
2317:(
2294:X
2253:H
2250:=
2247:H
2241:H
2238:=
2233:2
2229:H
2203:1
2187:T
2181:X
2174:1
2166:)
2161:X
2155:1
2139:T
2133:X
2127:(
2121:X
2117:=
2113:H
2097:.
2084:y
2078:1
2062:T
2056:X
2049:1
2041:)
2036:X
2030:1
2014:T
2008:X
2002:(
1997:=
1963:Σ
1927:.
1921:T
1915:X
1908:1
1900:)
1895:X
1888:T
1882:X
1876:(
1870:X
1862:P
1835:.
1831:y
1824:T
1818:X
1811:1
1803:)
1798:X
1791:T
1785:X
1779:(
1773:X
1769:=
1753:X
1749:=
1739:y
1709:,
1705:y
1698:T
1692:X
1685:1
1677:)
1672:X
1665:T
1659:X
1653:(
1648:=
1577:ε
1571:β
1547:X
1523:,
1515:+
1506:X
1502:=
1498:y
1463:T
1457:A
1450:1
1442:)
1437:A
1430:T
1424:A
1418:(
1412:A
1390:A
1368:x
1346:b
1324:A
1302:x
1299:A
1286:.
1269:b
1262:T
1256:A
1249:1
1241:)
1236:A
1229:T
1223:A
1217:(
1212:=
1204:x
1189:x
1186:A
1179:T
1173:A
1168:=
1160:b
1153:T
1147:A
1132:0
1129:=
1125:x
1122:A
1115:T
1109:A
1096:b
1089:T
1083:A
1060:.
1048:0
1045:=
1042:)
1038:x
1035:A
1027:b
1023:(
1017:T
1011:A
985:A
963:x
960:A
938:A
916:b
891:x
869:A
847:b
825:A
803:.
789:2
780:)
775:P
767:I
762:(
758:=
752:r
720:I
714:2
706:=
659:,
646:)
641:P
633:I
628:(
616:T
610:)
605:P
597:I
592:(
587:=
581:r
545:r
508:P
500:I
492:M
466:I
442:.
438:y
433:)
428:P
420:I
415:(
411:=
407:y
402:P
394:y
390:=
381:y
371:y
367:=
363:r
338:r
305:y
275:P
248:y
222:.
218:y
213:P
209:=
200:y
168:y
144:y
95:)
91:H
87:(
60:)
56:P
52:(
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
