equivalence transformation

equivalence transformation

[i′kwiv·ə·ləns ‚tranz·fər‚mā·shən]
(mathematics)
A mapping which associates with each square matrix A the matrix B = SAT, where S and T are nonsingular matrices. Also known as equivalent transformation.
References in periodicals archive ?
1 preserves the structure of the pencil, but note that in the real case or in the complex case with * being the complex conjugate, this transformation is a congruence transformation, while in the complex case with * being the transpose, this is just a structure preserving equivalence transformation but not a congruence transformation.
In this section we present several numerical tests to compare the computed finite eigenvalues of the subpencils generated by the three methods: structured unitary equivalence transformation (Algorithms 1 and 2), Schur complement transformation (by using (2.
max]: the maximum relative error of the finite eigenvalues for a given pencil with the structured unitary equivalence transformation method, the Schur complement method, and the non-structured equivalence transformation method, respectively.
min]: the minimum relative error of the finite eigenvalues for a given pencil with the structured unitary equivalence transformation method, the Schur complement method, and the non-structured equivalence transformation method, respectively.
GM]: the geometric mean of the relative errors of the finite eigenvalues for a given pencil with the structured unitary equivalence transformation method, the Schur complement method, and the non-structured equivalence transformation method, respectively.
hat]: the maximum real part of the eigenvalues computed by the non-structured equivalence transformation method in the case when all the exact finite eigenvalues are purely imaginary.
In this paper we develop numerical methods to calculate a structure preserving deflation under unitary (or in the real case real orthogonal) equivalence transformations for the infinite eigenvalue part in eigenvalue problems for structured matrix pencils