On the class of \(D_{k}\)-symmetrizable matrices
An \(n\times n\) real matrix \(X\) is said to be \(D_{k}\)-symmetrizable if there exists a nonsingular diagonal matrix \(D\) with \(s(D)=k\) and \(DX=X^{\prime }D\) , where \(X^{\prime }\) is the transpose of \(X\) and the signature \(s(D)\)\ is the absolute value of the difference of the number of positive eigenvalues and the number of negative eigenvalues. In this paper, the authors characterize the matrices \(D_{k}\)-symmetrizable proving that they have many properties similar to real symmetric matrices. They prove that if \(A\) is \(D_{k}\)-symmetrizable then, if \(k=n\), all the eigenvalues of \(A\) are real and \(s(A)=s(DA)\), and, if \(k<n\), \(A\) has at least \(k\) real eigenvalues. It is known that for every \(n\times n\) real matrix \(X\) there is a \(n\times n\) symmetric and nonsingular matrix \(S\) such that \(SX=X^{\prime }S\) with \(s(S)=k \), that is, \(X\) is \(S_{k}\)-symmetrizable. Then it is equivalent that \(A\) is \( S_{k}\)-symmetrizable and that there is an orthogonal \(Q\) such that \( Q^{\prime }AQ\) is \(D_{k}\)-symmetrizable. Moreover, they also prove that for any \(n\times n\) real matrix \(B\) there exist orthogonal \(P\) and \(Q\) such that \(PBQ\) is sign-symmetric (\( \text{sign}(x_{ij})= \text{sign}(x_{ji})\)) and \(PQ\) is a diagonal matrix with entries \(\pm 1 \), generalizing, in some sense, the singular value decomposition for real square matrices.
scientific article; zbMATH DE number 1903564