WDgeev WDsbev WDstev WDsyev

WDgeev() Function WDgeev ( JobVl As  String, JobVr As  String, N As  Long, A As  Variant  ) Eigenvalues and left and/or right eigenvectors of a general matrix Purpose WDgeev computes for an N x N real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. The right eigenvector v(j) of A satisfies A * v(j) = λ(j) * v(j) where λ(j) is its eigenvalue. The left eigenvector u(j) of A satisfies u(j)^H * A = λ(j) * u(j)^H where u(j)^H denotes the conjugate transpose of u(j). The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real. Returns N+1 x 2 (JobVl = "N", JobVr = "N") Column 1 Column 2 Rows 1 to N Real part of the computed eigenvalues Imaginary part of the computed eigenvalues Row N+1 Return code 0 N+1 x N+2 (JobVl = "V", JobVr = "N") Column 1 Column 2 Column 3 to N+2 Rows 1 to N Real part of the computed eigenvalues Imaginary part of the computed eigenvalues Left eigenvectors are stored one after another in the same order as their eigenvalues. If j-th eigenvalue is real number, the eigenvector is also real vector and stored in j-th column. If j-th and (j+1)-st eigenvalues form a complex conjugate pair, j-th column contains the real part, and (j+1)-st column contains the imaginary part. Row N+1 Return code 0 0 N+1 x N+2 (JobVl = "N", JobVr = "V") Column 1 Column 2 Column 3 to N+2 Rows 1 to N Real part of the computed eigenvalues Imaginary part of the computed eigenvalues Right eigenvectors are stored one after another in the same order as their eigenvalues. If j-th eigenvalue is real number, the eigenvector is also real vector and stored in j-th column. If j-th and (j+1)-st eigenvalues form a complex conjugate pair, j-th column contains the real part, and (j+1)-st column contains the imaginary part. Row N+1 Return code 0 0 N+1 x 2N+2 (JobVl = "V", JobVr = "V") Column 1 Column 2 Column 3 to N+2 Column N+3 to 2N+2 Rows 1 to N Real part of the computed eigenvalues Imaginary part of the computed eigenvalues Left eigenvectors are stored one after another in the same order as their eigenvalues. If j-th eigenvalue is real number, the eigenvector is also real vector and stored in j-th column. If j-th and (j+1)-st eigenvalues form a complex conjugate pair, j-th column contains the real part, and (j+1)-st column contains the imaginary part. Right eigenvectors are stored one after another in the same order as their eigenvalues. If j-th eigenvalue is real number, the eigenvector is also real vector and stored in j-th column. If j-th and (j+1)-st eigenvalues form a complex conjugate pair, j-th column contains the real part, and (j+1)-st column contains the imaginary part. Row N+1 Return code 0 0 0 Return code. = 0: Successful exit. = i > 0: Failed to converge for first to i-th eigenvalues, and no eigenvectors have been computed. Parameters [in] JobVl = "N": Left eigenvectors of A are not computed. = "V": Left eigenvectors of A are computed. [in] JobVr = "N": Right eigenvectors of A are not computed. = "V": Right eigenvectors of A are computed. [in] N Order of the matrix A. (N >= 1) [in] A (N x N) N x N matrix A. Reference LAPACK Example Compute all eigenvalues and eigenvectors of the general matrix A, where ( 0.20 -0.11 -0.93 ) A = ( -0.32 0.81 0.37 ) ( -0.80 -0.92 -0.29 )