GNU Scientific Library – Reference Manual: Tridiagonal Decomposition of Hermitian Matrices

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### 14.10 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 unitary matrix U. 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 * U, 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.