Eigenvalue bounds and inequalities using vector aggregation of matrices (Q1379086)

Let \((A_{ij})\in{\mathbb{C}}^{m\times m}\) be a block matrix with \(A_{ij}\in{\mathbb{C}}^{n_i\times n_j}\), \(i,j=1,\ldots,n\) and let \(X=(X_i)\in{\mathbb{C}}^{m\times k}\), (\(k<m\)) be consistently partitioned: \(X_i\in{\mathbb{C}}^{n_i\times k}\). If \(\text{ rank} X_i=k\) for all \(i\), then \(X\) is nondegenerately partitioned. The (nondegenerate) (block-)vector aggregated matrix \(A_X\) is a ``compressed'' matrix defined as \(A_X=((A_X)_{ij})\in{\mathbb{C}}^{nk\times nk}\) with \((A_X)_{ij}=P_i^* A_{ij}P_j\) where \(P_i=X_i(X_i^*X_i)^{-1/2}\). In this paper the relation between the eigenvalues of \(A\) and \(A_X\), and many of its consequences, is investigated. Among other things it is proved that for \(A\) Hermitian, a generalized interlacing property for the eigenvalues \(A\) and \(A_X\) holds. This implies bounds for the eigenvalues of \(A\) in terms of the eigenvalues of its diagonal blocks \(A_{ii}\). Also, lower bounds for the spectral parameters \(\xi_i(A)=\lambda_{\max}\{(A)_{ii}(A^{-1})_{ii}\}\) of a positive definite \(A\) are derived. The \(\xi_i(A)\) measure in a sense the distance of \(A\) from its block diagonal. Furthermore, it is proved that for a normal matrix \(A\) and in the case \(k=1\), the eigenvalue of \(A\) which is closest to an arbitrary \(\xi\in{\mathbb{C}}\) lies in a disk with center \(\xi\) and with radius \(\lambda_{\min}\{[(A-\xi I)^*(A-\xi I)]_X\}\).