=============================== Rectangular matrix =============================== R: 1,4000 2,5000 3,6000 4,9000 3,0000 4,6000 9,9000 3,8000 5,7000 10,4500 8,0400 7,5000 ----- Computes Singular Value Decomposition Jacobi method ----- rank: 3 Singular values: 20,9345 3,0672 1,6466 Condition number: 12,7137 ---- Computes Singular Value Decomposition ----- Q 20,9345 0,0000 0,0000 0,0000 3,0672 0,0000 0,0000 0,0000 1,6466 U 0,1922 0,6787 -0,4864 0,3488 0,1373 -0,4989 0,5661 -0,6667 -0,3356 0,7217 0,2758 0,6339 V 0,7225 -0,6831 0,1070 0,4529 0,5845 0,6732 0,5224 0,4379 -0,7316 Verify U*Q*V'=R 1,4000 2,5000 3,6000 4,9000 3,0000 4,6000 9,9000 3,8000 5,7000 10,4500 8,0400 7,5000 ----- Computes left pseudo-inverse matrix using QR decomposition ----- Left pseudo Inverse: -0,1761 -0,0510 0,1462 0,0047 -0,0654 -0,1703 -0,2520 0,3274 0,3178 0,2500 0,0681 -0,2243 computes pinv(R)*R 1,0000 0,0000 0,0000 0,0000 1,0000 -0,0000 0,0000 -0,0000 1,0000 ----- Computes right pseudo-inverse matrix using SVD ----- Right pseudo Inverse: -0,1761 -0,0654 0,3178 -0,0510 -0,1703 0,2500 0,1462 -0,2520 0,0681 0,0047 0,3274 -0,2243 computes R*pinv(R) 1,0000 0,0000 0,0000 0,0000 1,0000 -0,0000 0,0000 0,0000 1,0000 =============================== Square matrix =============================== S: 0,0000 0,0000 1,0000 0,0000 0,0000 0,0000 0,0000 1,0000 -3,0000 0,0000 -5,0000 0,0000 0,0000 -4,0000 0,0000 -6,0000 ----- Computes inverse matrix using Java3D inner function ----- Inverse: -1,6667 -0,0000 -0,3333 -0,0000 -0,0000 -1,5000 -0,0000 -0,2500 1,0000 0,0000 0,0000 0,0000 0,0000 1,0000 0,0000 0,0000 ----- Computes inverse matrix using LU decomposition ----- Inverse: -1,6667 -0,0000 -0,3333 -0,0000 -0,0000 -1,5000 -0,0000 -0,2500 1,0000 0,0000 0,0000 0,0000 0,0000 1,0000 0,0000 0,0000 ----- Computes Eigen value decomposition: Matrix of eigenvalues -0,6972 0,0000 0,0000 0,0000 0,0000 -4,3028 0,0000 0,0000 0,0000 0,0000 -0,7639 0,0000 0,0000 0,0000 0,0000 -5,2361 Matrix of eigenvectors 0,8203 0,3381 0,0000 0,0000 0,0000 0,0000 -0,7947 -0,2814 -0,5719 -1,4548 0,0000 0,0000 0,0000 0,0000 0,6071 1,4734 Real part of eigenvalues -0,6972 -4,3028 -0,7639 -5,2361 Imaginary part of eigenvalues 0,0000 0,0000 0,0000 0,0000 Verify V*D*inv(V)=S 0,0000 0,0000 1,0000 0,0000 0,0000 0,0000 0,0000 1,0000 -3,0000 0,0000 -5,0000 0,0000 0,0000 -4,0000 0,0000 -6,0000 The matrix is DEFINTE NEGATIVE ----- Computes solution of Lyapunov equation S*X + X*S' = -C: With C= 1,0000 0,0000 0,0000 0,0000 0,0000 1,0000 0,0000 0,0000 0,0000 0,0000 1,0000 0,0000 0,0000 0,0000 0,0000 1,0000 Solution X= 0,9667 -0,0000 -0,5000 -0,0000 -0,0000 0,8542 -0,0000 -0,5000 -0,5000 -0,0000 0,4000 -0,0000 -0,0000 -0,5000 -0,0000 0,4167 The solution X is SYMMETRIC and DEFINITE POSITIVE Verify S*X + X*S' = -C: -1,0000 0,0000 0,0000 0,0000 0,0000 -1,0000 0,0000 0,0000 0,0000 0,0000 -1,0000 0,0000 0,0000 0,0000 0,0000 -1,0000 =============================== Square quasi singular matrix =============================== S: 0,0000 0,0000 1,0000 0,0000 0,0000 -2,0000 0,0000 1,0000 -3,0000 0,0000 -5,0000 0,0000 3,0200 -4,1200 8,9600 1,9920 ----- Computes Singular Value Decomposition ----- rank: 4 Singular values: 11,8136 3,1768 0,8313 0,0131 Condition number: 903,3350 ----- Computes damped least square inverse matrix using SVD decomposition ----- inverse (eps=0): -1,6667 0,0000 -0,3333 -0,0000 28,8725 14,6471 -7,4020 -7,3529 1,0000 -0,0000 0,0000 0,0000 57,7451 30,2941 -14,8039 -14,7059 computes inv(S)*S 1,0000 0,0000 -0,0000 -0,0000 0,0000 1,0000 0,0000 0,0000 0,0000 -0,0000 1,0000 0,0000 0,0000 -0,0000 0,0000 1,0000 inverse (eps=0.02, lambdaMax=0.002): -1,6518 0,0078 -0,3371 -0,0038 28,4930 14,4494 -7,3048 -7,2563 0,9906 -0,0049 0,0024 0,0024 56,9811 29,8962 -14,6082 -14,5114 computes inv(S)*S 1,0000 0,0001 0,0000 0,0002 0,0001 0,9974 -0,0001 -0,0053 0,0000 -0,0001 1,0000 -0,0001 0,0002 -0,0053 -0,0001 0,9894 inverse (eps=0.02, lambdaMax=0.02): -1,0207 0,3361 -0,4986 -0,1643 12,4270 6,0815 -3,1900 -3,1674 0,5915 -0,2126 0,1045 0,1039 24,6413 13,0520 -6,3254 -6,2806 computes inv(S)*S 0,9996 0,0045 0,0002 0,0089 0,0045 0,8867 -0,0028 -0,2280 0,0002 -0,0028 0,9999 -0,0056 0,0089 -0,2280 -0,0056 0,5411 inverse (eps=0.02, lambdaMax=0.2): -0,5293 0,5623 -0,6029 -0,2760 0,3521 -0,2004 -0,1020 -0,0971 0,2830 -0,3581 0,1729 0,1758 0,3461 0,3935 -0,1002 -0,0954 computes inv(S)*S 0,9752 0,0127 0,0121 0,0124 0,0127 0,8008 -0,0081 -0,3938 0,0121 -0,0081 0,9935 -0,0079 0,0124 -0,3938 -0,0079 0,2034