Study of weighted pseudoinverse matrices with in definite singular weights
DOI:
https://doi.org/10.15407/dopovidi2022.06.017Keywords:
weighted pseudoinverse matrices with singular indefinite weights, matrix power series, matrix power products, singular indefinite weights, expansions of the weighted pseudoinverse matricesAbstract
The weighted pseudoinverse matrix with singular indefinite weights is investigated in the paper. The weighted matrix norms with indefinite weights are specified, and inequalities for norms of matrix products are established. It is shown that under certain conditions a matrix symmetrized from the left by a positive semidefinite symmetrizer [symmetrization operator] can be diagonalized by means of weighted orthogonal transformation. The necessary and sufficient conditions for the existence of the version under consideration of pseudoinverse matrices with singular indefinite weights are specified. And on basis of a theorem due to Cayley-Hamilton the representation of the weighted pseudoinverse matrix with indefinite singular weights is obtained in terms of coefficients of characteristic polynomials of symmetrizable matrices. The expansions of weighted pseudoinverse matrices with positive semidefinite and indefinite singular weights in matrix power series and matrix products are derived and investigated based on characteristics of symmetrizable matrices as well as on the representation of pseudoinverse matrices in terms of coefficients of characteristic polynomials of symmetrizable matrices. On the basis of these expansions the limitary representations of weighted pseudoinverse matrices with these weights are obtained.
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