A = hilb(10); n = 10; b = A * ones(n,1); b = b + 1.e-9*randn(n,1); A\b ans = 1.0e+04 * 0.0001 -0.0002 0.0059 -0.0529 0.2542 -0.7011 1.1543 -1.1184 0.5887 -0.1296 help pinv PINV Pseudoinverse. X = PINV(A) produces a matrix X of the same dimensions as A' so that A*X*A = A, X*A*X = X and A*X and X*A are Hermitian. The computation is based on SVD(A) and any singular values less than a tolerance are treated as zero. PINV(A,TOL) treats all singular values of A that are less than TOL as zero. By default, TOL = max(size(A)) * eps(norm(A)). Class support for input A: float: double, single See also RANK. Documentation for pinv doc pinv Other functions named pinv sym/pinv x = pinv(A,1.e-08)*b x = 1.0000 1.0003 0.9979 1.0046 0.9982 0.9965 0.9997 1.0031 1.0028 0.9969 x = pinv(A,1.e-07)*b x = 1.0000 1.0003 0.9979 1.0046 0.9982 0.9965 0.9997 1.0031 1.0028 0.9969 x = pinv(A,1.e-04)*b x = 1.0000 0.9991 1.0032 0.9977 0.9972 0.9998 1.0023 1.0030 1.0010 0.9965 x = pinv(A,1.e-02)*b x = 1.0150 0.9233 1.0130 1.0541 1.0587 1.0428 1.0163 0.9846 0.9509 0.9169 x = pinv(A,1.e-02\4)*b x = 0 0 0 0 0 0 0 0 0 0 x = pinv(A,1.e-04)*b x = 1.0000 0.9991 1.0032 0.9977 0.9972 0.9998 1.0023 1.0030 1.0010 0.9965 trigtest1 addpath ../APPL/ mir ans = 256 256 .computing the T-SVD solution --- wait.... done. close all close all mir ans = 256 256 .computing the T-SVD solution --- wait.... done. close close all %% solve LS problem A = randn(7,3); b = randn(7,1); %% min || b - Ax||_2 x? x = (A'*A) \ (A'*b) x = -0.0452 -0.3616 0.2044 %% it is also the Pseudo inverse solution/ irDat A = 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 1 1 0 0 1 0 0 0 0 size(A) ans = 9 8 q q = 0 0.7071 0 0 0 0 0 0.7071 0 s0 = q'*A s0 = Columns 1 through 5 0 0.4082 0.8165 0.3162 0 Columns 6 through 8 0 0 0.6325 [U S V] = svd(A); B = U(:,1:3)*S(1:3,1:3)*(V(:,1:3))' B = Columns 1 through 5 0.2561 0.2486 0.0738 0.4055 0.4687 -0.0746 0.4758 0.6186 0.0830 0.0084 -0.1214 0.1477 -0.0335 0.0243 0.4614 0.2074 0.0424 0.0222 0.2131 0.0435 -0.0746 0.4758 0.6186 0.0830 0.0084 0.5578 0.0437 -0.0071 0.5411 0.0674 -0.1214 0.1477 -0.0335 0.0243 0.4614 0.1319 0.3593 0.4438 0.2412 0.0326 0.5578 0.0437 -0.0071 0.5411 0.0674 Columns 6 through 8 0.3957 0.4687 0.3074 -0.0169 0.0084 0.4684 0.5272 0.4614 0.3075 -0.0279 0.0435 -0.0005 -0.0169 0.0084 0.4684 -0.1214 0.0674 -0.0822 0.5272 0.4614 0.3075 -0.0483 0.0326 0.3132 -0.1214 0.0674 -0.0822 B = B ./ sqrt( sum(A .^2, 1)); norm(B(:,4)) ans = 0.9319 B = U(:,1:3)*S(1:3,1:3)*(V(:,1:3))' B = Columns 1 through 5 0.2561 0.2486 0.0738 0.4055 0.4687 -0.0746 0.4758 0.6186 0.0830 0.0084 -0.1214 0.1477 -0.0335 0.0243 0.4614 0.2074 0.0424 0.0222 0.2131 0.0435 -0.0746 0.4758 0.6186 0.0830 0.0084 0.5578 0.0437 -0.0071 0.5411 0.0674 -0.1214 0.1477 -0.0335 0.0243 0.4614 0.1319 0.3593 0.4438 0.2412 0.0326 0.5578 0.0437 -0.0071 0.5411 0.0674 Columns 6 through 8 0.3957 0.4687 0.3074 -0.0169 0.0084 0.4684 0.5272 0.4614 0.3075 -0.0279 0.0435 -0.0005 -0.0169 0.0084 0.4684 -0.1214 0.0674 -0.0822 0.5272 0.4614 0.3075 -0.0483 0.0326 0.3132 -0.1214 0.0674 -0.0822 B = U(:,1:3)*S(1:3,1:3)*(V(:,1:3))'; B = B ./ sqrt( sum(B .^2, 1)); norm(B(:,4)) ans = 1.0000 s1 = q'*B s1 = Columns 1 through 5 0.0456 0.7094 0.7625 0.2460 0.0357 Columns 6 through 8 -0.0534 0.0357 0.6052 s0 s0 = Columns 1 through 5 0 0.4082 0.8165 0.3162 0 Columns 6 through 8 0 0 0.6325 diary off