我正在使用 Eigen 将代码从 Matlab 移植到 C++，但我发现了一些问题。我需要计算 QR 分解，
代码matlab：
[Q,R] = qr(Xtilde, 0);
代码特征：
HouseholderQR<MatrixXd> qr(X_TILDE.rows(), X_TILDE.cols());
qr.compute(X_TILDE);
MatrixXd q = qr.householderQ()*MatrixXd::Identity(X_TILDE.rows(), X_TILDE.cols());
MatrixXd temp = qr.matrixQR().triangularView<Upper>();
MatrixXd r = temp.topRows(X_TILDE.cols());
Xtilde(46 x 4) 是：
0.0577350269189626 -0.0823586409336270 0.117484066404866 -0.167590257713776
0.0577350269189626 -0.0776524328802769 0.104440937399917 -0.140470929246880
0.0577350269189626 -0.0729462248269268 0.0921650512776130 -0.116447378835011
0.0577350269189626 -0.0682400167735767 0.0806564080379527 -0.0953319835657442
0.0577350269189626 -0.0635338087202266 0.0699150076809364 -0.0769371205266531
0.0577350269189626 -0.0588276006668765 0.0599408502065641 -0.0610751668053119
0.0577350269189626 -0.0541213926135263 0.0507339356148359 -0.0475584994892948
0.0577350269189626 -0.0494151845601762 0.0422942639057516 -0.0361994956661756
0.0577350269189626 -0.0447089765068261 0.0346218350793114 -0.0268105324235286
0.0577350269189626 -0.0400027684534760 0.0277166491355153 -0.0192039868489278
0.0577350269189626 -0.0352965604001259 0.0215787060743631 -0.0131922360299474
0.0577350269189626 -0.0305903523467758 0.0162080058958549 -0.00858765705416130
0.0577350269189626 -0.0258841442934256 0.0116045485999908 -0.00520262700914369
0.0577350269189626 -0.0211779362400755 0.00776833418677071 -0.00284952298246863
0.0577350269189626 -0.0164717281867254 0.00469936265619463 -0.00134072206171021
0.0577350269189626 -0.0117655201333753 0.00239763400826257 -0.000488601334442496
0.0577350269189626 -0.00705931208002518 0.000863148242974523 -0.000105537888239579
0.0577350269189626 -0.00235310402667506 9.59053603305026e-05 -3.90881067553996e-06
0.0577350269189626 0.00235310402667506 9.59053603305026e-05 3.90881067553996e-06
0.0577350269189626 0.00705931208002518 0.000863148242974523 0.000105537888239579
0.0577350269189626 0.0117655201333753 0.00239763400826257 0.000488601334442496
0.0577350269189626 0.0164717281867254 0.00469936265619463 0.00134072206171021
0.0577350269189626 0.0211779362400755 0.00776833418677071 0.00284952298246863
0.0577350269189626 0.0258841442934256 0.0116045485999908 0.00520262700914369
0.0577350269189626 0.0305903523467758 0.0162080058958549 0.00858765705416130
0.0577350269189626 0.0352965604001259 0.0215787060743631 0.0131922360299474
0.0577350269189626 0.0400027684534760 0.0277166491355153 0.0192039868489278
0.0577350269189626 0.0447089765068261 0.0346218350793114 0.0268105324235286
0.0577350269189626 0.0494151845601762 0.0422942639057516 0.0361994956661756
0.0577350269189626 0.0541213926135263 0.0507339356148359 0.0475584994892948
0.0577350269189626 0.0588276006668765 0.0599408502065641 0.0610751668053119
0.0577350269189626 0.0635338087202266 0.0699150076809364 0.0769371205266531
0.0577350269189626 0.0682400167735767 0.0806564080379527 0.0953319835657442
0.0577350269189626 0.0729462248269268 0.0921650512776130 0.116447378835011
0.0577350269189626 0.0776524328802769 0.104440937399917 0.140470929246880
0.0577350269189626 0.0823586409336270 0.117484066404866 0.167590257713776
0.0577350269189626 0.0870648489869772 0.131294438292458 0.197992987148126
0.0577350269189626 0.0917710570403273 0.145872053062695 0.231866740462355
0.0577350269189626 0.0964772650936774 0.161216910715575 0.269399140568890
1.0000e-05 0 0 0
0 0.00100000000000000 0 0
0 0 0.00100000000000000 0
0 0 0 0.00100000000000000
当我执行 QR 分解时，我从 Eigen 获得的 Q 矩阵与我从 Matlab 获得的 Q 矩阵不同。此外，来自 Eigen 的 R 矩阵与我从 Matlab 获得的 Q 矩阵具有不同的符号：来自 Matlab 的 Q
-0.154303349906983 -0.260970906342325 -0.321232647415310 0.353839296299944
-0.154303349906984 -0.248240618228066 -0.274222991714045 0.250277645081041
-0.154303349906984 -0.235510330113806 -0.229563818797843 0.159661173523266
-0.154303349906984 -0.222780041999546 -0.187255128666704 0.0813260262593829
-0.154303349906984 -0.210049753885286 -0.147296921320628 0.0146083479221537
-0.154303349906984 -0.197319465771027 -0.109689196759616 -0.0411557168556593
-0.154303349906984 -0.184589177656767 -0.0744319549836670 -0.0866300234412932
-0.154303349906984 -0.171858889542507 -0.0415251959927814 -0.122478427201985
-0.154303349906984 -0.159128601428247 -0.0109689197869589 -0.149364783504973
-0.154303349906984 -0.146398313313987 0.0172368736338002 -0.167952947717494
-0.154303349906984 -0.133668025199728 0.0430921842694962 -0.178906775206785
-0.154303349906984 -0.120937737085468 0.0665970121201288 -0.182890121340084
-0.154303349906984 -0.108207448971208 0.0877513571856982 -0.180566841484628
-0.154303349906984 -0.0954771608569483 0.106555219466204 -0.172600791007654
-0.154303349906984 -0.0827468727426885 0.123008598961647 -0.159655825276400
-0.154303349906984 -0.0700165846284287 0.137111495672027 -0.142395799658103
-0.154303349906984 -0.0572862965141690 0.148863909597343 -0.121484569520001
-0.154303349906984 -0.0445560083999092 0.158265840737596 -0.0975859902293308
-0.154303349906984 -0.0318257202856494 0.165317289092786 -0.0713639171533296
-0.154303349906984 -0.0190954321713897 0.170018254662912 -0.0434822056592351
-0.154303349906984 -0.00636514405712988 0.172368737447976 -0.0146047111142846
-0.154303349906984 0.00636514405712989 0.172368737447976 0.0146047111142846
-0.154303349906984 0.0190954321713897 0.170018254662912 0.0434822056592352
-0.154303349906984 0.0318257202856494 0.165317289092786 0.0713639171533297
-0.154303349906984 0.0445560083999092 0.158265840737596 0.0975859902293308
-0.154303349906984 0.0572862965141690 0.148863909597343 0.121484569520001
-0.154303349906984 0.0700165846284287 0.137111495672027 0.142395799658103
-0.154303349906984 0.0827468727426885 0.123008598961647 0.159655825276400
-0.154303349906984 0.0954771608569483 0.106555219466204 0.172600791007654
-0.154303349906984 0.108207448971208 0.0877513571856981 0.180566841484628
-0.154303349906984 0.120937737085468 0.0665970121201288 0.182890121340084
-0.154303349906984 0.133668025199728 0.0430921842694961 0.178906775206785
-0.154303349906984 0.146398313313987 0.0172368736338002 0.167952947717494
-0.154303349906984 0.159128601428247 -0.0109689197869590 0.149364783504973
-0.154303349906984 0.171858889542507 -0.0415251959927814 0.122478427201985
-0.154303349906984 0.184589177656767 -0.0744319549836671 0.0866300234412932
-0.154303349906984 0.197319465771026 -0.109689196759616 0.0411557168556592
-0.154303349906984 0.210049753885286 -0.147296921320628 -0.0146083479221538
-0.154303349906984 0.222780041999546 -0.187255128666704 -0.0813260262593831
-0.154303349906984 0.235510330113806 -0.229563818797843 -0.159661173523266
-0.154303349906984 0.248240618228066 -0.274222991714045 -0.250277645081041
-0.154303349906984 0.260970906342325 -0.321232647415310 -0.353839296299943
-2.67261241816974e-05 -8.58599617264519e-22 2.99060305347150e-05 1.01643998560347e-21
0 0.00270499900768257 -2.92157489996581e-19 0.00621245013765559
0 0 -0.00306354459354929 -4.46671225035349e-19
0 0 0 -0.00353824252408127
来自 Matlab 的 R
-0.374165738811025 -1.38777878078145e-17 -0.365257030483198 0
0 0.369685902715625 -2.77555756156289e-17 0.649094916921038
0 0 -0.326419273316809 3.81639164714898e-17
0 0 0 -0.282626188904237
来自本征的 Q
-0.154303 -0.260971 -0.321233 0.35384
-0.154303 -0.248241 -0.274223 0.250278
-0.154303 -0.23551 -0.229564 0.159661
-0.154303 -0.22278 -0.187255 0.081326
-0.154303 -0.21005 -0.147297 0.0146083
-0.154303 -0.197319 -0.109689 -0.0411558
-0.154303 -0.184589 -0.074432 -0.0866302
-0.154303 -0.171859 -0.0415252 -0.122479
-0.154303 -0.159129 -0.0109689 -0.149365
-0.154303 -0.146398 0.0172369 -0.167953
-0.154303 -0.133668 0.0430922 -0.178907
-0.154303 -0.120938 0.066597 -0.18289
-0.154303 -0.108207 0.0877514 -0.180567
-0.154303 -0.0954772 0.106555 -0.172601
-0.154303 -0.0827469 0.123009 -0.159656
-0.154303 -0.0700166 0.137112 -0.142396
-0.154303 -0.0572863 0.148864 -0.121485
-0.154303 -0.044556 0.158266 -0.0975861
-0.154303 -0.0318257 0.165317 -0.071364
-0.154303 -0.0190954 0.170018 -0.0434823
-0.154303 -0.00636514 0.172369 -0.0146047
-0.154303 0.00636514 0.172369 0.0146047
-0.154303 0.0190954 0.170018 0.0434823
-0.154303 0.0318257 0.165317 0.071364
-0.154303 0.044556 0.158266 0.0975861
-0.154303 0.0572863 0.148864 0.121485
-0.154303 0.0700166 0.137112 0.142396
-0.154303 0.0827469 0.123009 0.159656
-0.154303 0.0954772 0.106555 0.172601
-0.154303 0.108207 0.0877514 0.180567
-0.154303 0.120938 0.066597 0.18289
-0.154303 0.133668 0.0430922 0.178907
-0.154303 0.146398 0.0172369 0.167953
-0.154303 0.159129 -0.0109689 0.149365
-0.154303 0.171859 -0.0415252 0.122479
-0.154303 0.184589 -0.074432 0.0866302
-0.154303 0.197319 -0.109689 0.0411558
-0.154303 0.21005 -0.147297 -0.0146083
-0.154303 0.22278 -0.187255 -0.081326
-0.154303 0.23551 -0.229564 -0.159661
-0.154303 0.248241 -0.274223 -0.250278
-0.154303 0.260971 -0.321233 -0.35384
-2.67261e-05 3.38813e-21 2.9906e-05 -1.35525e-20
0 0.0026726 4.33681e-19 0.00613805
0 0 -0.0029906 1.02999e-18
0 0 0 -0.00341263
来自本征的 R
-0.374166 1.38778e-17 -0.374166 0
0 0.374167 2.77556e-17 0.672987
0 0 -0.334381 -6.245e-17
0 0 0 -0.293029
我不明白为什么我在Q矩阵的最后四行有不同，为什么我在Q矩阵中有不同的符号，
我获得 Q'*Q ： Matlab ：
1.0000000 7.979727989e-17 9.012480877e-17 6.938893903e-18
7.979727989e-17 1.0000000 -3.469446951e-17 3.093533729e-17
9.012480877e-17 -3.469446951e-17 1.000000 1.387915620e-17
6.938893903e-18 3.093533729e-17 1.387915620e-17 1.000000
特征：
1 -1.8735e-16 -1.09631e-16 5.55112e-17
-1.8735e-16 1 -1.52654e-16 1.31595e-17
-1.09631e-16 -1.52654e-16 1 1.94291e-16
5.55112e-17 1.31595e-17 1.94291e-16 1
对于 Q*R - X：
特征：
5.55112e-17 1.38778e-17 -1.66533e-16 5.55112e-17
0 5.55112e-17 -5.55112e-17 1.11022e-16
0 0 -2.77556e-17 2.77556e-17
0 2.77556e-17 -2.77556e-17 8.32667e-17
0 0 -2.77556e-17 2.77556e-17
0 1.38778e-17 -2.77556e-17 4.16334e-17
0 0 -2.77556e-17 1.38778e-17
0 0 -2.77556e-17 0
0 6.93889e-18 -2.08167e-17 1.38778e-17
0 6.93889e-18 -2.08167e-17 0
0 6.93889e-18 -2.08167e-17 4.85723e-17
0 1.38778e-17 -6.93889e-18 1.38778e-17
0 6.93889e-18 -2.08167e-17 2.42861e-17
0 6.93889e-18 -3.1225e-17 2.25514e-17
0 3.46945e-18 -1.04083e-17 2.08167e-17
0 3.46945e-18 -3.1225e-17 2.60209e-17
0 3.46945e-18 -7.80626e-18 1.38778e-17
0 0 -2.42861e-17 2.1684e-18
0 3.46945e-18 -2.60209e-17 1.31188e-17
0 3.46945e-18 1.0842e-19 5.81403e-18
0 3.46945e-18 -1.46367e-17 1.02703e-17
0 2.60209e-18 -1.46367e-17 1.87279e-18
0 5.20417e-18 -2.07083e-17 6.32903e-18
0 3.46945e-18 -5.20417e-18 -2.71051e-18
0 3.46945e-18 -2.42861e-17 1.30104e-18
0 3.46945e-18 -1.47451e-17 -6.93889e-18
0 -1.04083e-17 -1.73472e-17 -1.21431e-17
0 0 3.46945e-18 -6.93889e-18
0 0 -1.73472e-17 -1.56125e-17
0 -6.93889e-18 -1.73472e-17 -2.42861e-17
0 0 -2.08167e-17 -2.77556e-17
0 -6.93889e-18 -2.08167e-17 -2.08167e-17
0 1.38778e-17 -2.77556e-17 -3.46945e-17
0 -6.93889e-18 -2.08167e-17 -2.08167e-17
0 0 -1.38778e-17 -5.55112e-17
0 0 -1.38778e-17 -2.77556e-17
0 -1.38778e-17 -2.77556e-17 -1.38778e-17
0 -1.38778e-17 -2.77556e-17 0
0 1.38778e-17 -2.77556e-17 0
0 1.38778e-17 -2.77556e-17 -5.55112e-17
0 0 -2.77556e-17 0
0 1.38778e-17 0 0
0 2.55373e-22 -1.69407e-21 1.13989e-21
0 0 1.07448e-20 -6.50521e-19
0 0 0 2.32655e-20
0 0 0 0
MATLAB：
-3.469446951e-17 1.387778780e-17 5.551115123e-17 -5.55111512e-17
6.9388939039e-18 -1.38777878e-17 0 -1.11022302e-16
6.9388939039e-18 -1.38777878e-17 2.775557561e-17 -2.77555756e-17
6.9388939039e-18 -1.38777878e-17 1.387778780e-17 0
6.9388939039e-18 -1.38777878e-17 1.387778780e-17 -2.77555756e-17
6.9388939039e-18 -1.38777878e-17 1.387778780e-17 -4.16333632e-17
6.9388939039e-18 0 1.387778780e-17 -1.38777878e-17
6.9388939039e-18 0 1.387778780e-17 0
6.9388939039e-18 0 1.387778780e-17 -6.93889390e-18
6.9388939039e-18 0 2.081668171e-17 -2.77555756e-17
6.9388939039e-18 0 1.387778780e-17 0
6.9388939039e-18 0 1.387778780e-17 -3.46944695e-18
6.9388939039e-18 0 0 -1.04083408e-17
6.9388939039e-18 0 1.040834085e-17 -1.90819582e-17
6.9388939039e-18 0 0 0
6.9388939039e-18 3.4694469519e-18 1.734723475e-17 7.806255641e-18
6.9388939039e-18 0 9.540979117e-18 -3.035766082e-18
6.9388939039e-18 3.469446951e-18 7.806255641e-18 1.301042606e-18
6.9388939039e-18 1.734723475e-18 1.127570259e-17 -6.071532166e-18
6.9388939039e-18 2.602085213e-18 1.929879867e-17 -2.574980159e-18
6.9388939039e-18 3.035766082e-18 2.318837396e-17 -2.631731367e-18
6.9388939039e-18 2.602085213e-18 3.224146210e-17 -1.479766558e-18
6.9388939039e-18 4.336808689e-18 1.929879867e-17 -1.639855785e-18
6.9388939039e-18 3.469446951e-18 2.949029909e-17 -3.577867169e-18
6.9388939039e-18 3.469446951e-18 2.602085213e-17 4.3368086899e-19
6.9388939039e-18 -3.469446951e-18 2.775557561e-17 3.0357660829e-18
6.9388939039e-18 -3.469446951e-18 1.734723475e-17 -1.561251128e-17
6.9388939039e-18 3.469446951e-18 1.734723475e-17 -1.734723475e-18
6.9388939039e-18 6.938893903e-18 2.081668171e-17 1.7347234759e-18
6.9388939039e-18 6.938893903e-18 1.040834085e-17 1.0408340855e-17
6.9388939039e-18 0 1.387778780e-17 3.4694469519e-18
6.9388939039e-18 -6.93889390e-18 1.387778780e-17 -6.938893903e-18
6.9388939039e-18 0 1.387778780e-17 -2.081668171e-17
6.9388939039e-18 0 1.387778780e-17 1.387778780e-17
6.9388939039e-18 0 1.387778780e-17 0
6.9388939039e-18 0 1.387778780e-17 1.387778780e-17
6.9388939039e-18 0 1.387778780e-17 2.775557561e-17
6.9388939039e-18 -1.387778780e-17 1.387778780e-17 0
6.9388939039e-18 0 1.387778780e-17 2.775557561e-17
6.9388939039e-18 0 2.775557561e-17 2.775557561e-17
6.9388939039e-18 -1.387778780e-17 2.775557561e-17 0
6.9388939039e-18 1.387778780e-17 2.775557561e-17 5.5511151231e-17
0 5.348730573e-23 2.819782315e-21 2.967460447e-22
0 0 2.028703108e-20 1.791731434e-19
0 0 0 9.324126050e-21
0 0 0 0
如您所见，Q' Q 等价于单位矩阵，并且 Q R - X 非常接近于零。
此致，