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Perturbation and Location of the Singular Values of Symmetrically Scaled Matrices

In this report two new results on the relative perturbations of the singular values of square matrices are presented. The first one considers the scaled diagonally dominant matrices of the form $G=D^*BD$, where $D$ is diagonal and nonsingular. A simple and sharp estimate uses the norm of $B$ in the assumption and in the bound. The second result deals with a general square matrix and uses in the assumption and in the bound the scaled polar factor of the matrix. In addition a new location result for the singular values of a general matrix is presented. The intervals containing the singular values are defined using the absolute values of the diagonal elements of the triangular matrix $R$ which is obtained by the QR factorization with column pivoting of the original matrix. In the bounds for the intervals, the norms of some principal sub-matrices of $DRD$ are used, where $D$ is such that $|\diag (DRD)|=I$.

Singular values; Relative perturbations; Symmetric scaling; Location of the singular values

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