GNU Scientific Library Reference Manual - Third Edition (v1.12)by M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. Rossi Paperback (6"x9"), 592 pages, 60 figures ISBN 0954612078 RRP £24.95 ($39.95) |

## 13.7 Tridiagonal Decomposition of Hermitian Matrices

A hermitian matrix A can be factorized by similarity transformations into the form,

A = U T U^T

where U is a unitary matrix and T is a real symmetric tridiagonal matrix.

__Function:__int**gsl_linalg_hermtd_decomp***(gsl_matrix_complex **`A`, gsl_vector_complex *`tau`)- This function factorizes the hermitian matrix
`A`into the symmetric tridiagonal decomposition U T U^T. On output the real parts of the diagonal and subdiagonal part of the input matrix`A`contain the tridiagonal matrix T. The remaining lower triangular part of the input matrix contains the Householder vectors which, together with the Householder coefficients`tau`, encode the orthogonal matrix Q. This storage scheme is the same as used by lapack. The upper triangular part of`A`and imaginary parts of the diagonal are not referenced.

__Function:__int**gsl_linalg_hermtd_unpack***(const gsl_matrix_complex **`A`, const gsl_vector_complex *`tau`, gsl_matrix_complex *`Q`, gsl_vector *`diag`, gsl_vector *`subdiag`)- This function unpacks the encoded tridiagonal decomposition (
`A`,`tau`) obtained from`gsl_linalg_hermtd_decomp`

into the unitary matrix`U`, the real vector of diagonal elements`diag`and the real vector of subdiagonal elements`subdiag`.

__Function:__int**gsl_linalg_hermtd_unpack_T***(const gsl_matrix_complex **`A`, gsl_vector *`diag`, gsl_vector *`subdiag`)- This function unpacks the diagonal and subdiagonal of the encoded
tridiagonal decomposition (
`A`,`tau`) obtained from the`gsl_linalg_hermtd_decomp`

into the real vectors`diag`and`subdiag`.

ISBN 0954612078 | GNU Scientific Library Reference Manual - Third Edition (v1.12) | See the print edition |