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Scalings and Submatrices
ÕªÒª£º For a positive diagonal matrix $\mathbf D$ and a structured matrix $\mathbf A$, we study diagonal scalings of the form ${\mathbf D}{\mathbf A}$. In this talk, we consider the problem of characterizing the properties of (infinitely many) scalings ${\mathbf D}{\mathbf A}$ in terms of the properties of the principal submatrices of $\mathbf A$. Since the number of the principal submatrices of an $n \times n$ matrix $\mathbf A$ is finite, we finally obtain only a finite number of conditions. Our major interest is the case, when all the scalings ${\mathbf D}{\mathbf A}$ are $Q^2$-matrices, since this property leads to positive stability and $D$-stability of $P$-matrices. We introduce the property of $D_{\theta}$-stability, i.e., the stability with respect to a given order $\theta$. For an $n \times n$ $P$-matrix $\mathbf A$, we prove a new criterion of $D$-stability and $D_{\theta}$-stability, based on the properties of matrix scalings.