Block Diagonalization and Eigenvalues

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Block Diagonalization and Eigenvalues

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Title: Block Diagonalization and Eigenvalues
Author: Eisenfeld, Jerome
Abstract: Let A denote an Algebra with an identity element. Consider an [see pdf for notation] matrix [see pdf for notation] with a partitioning [see pdf for notation] where E and H have respective orders [see pdf for notation] and [see pdf for notation]. We seek to obtain conditions under which A is similar to a matrix D of the form [see pdf for notation] where [see pdf for notation] denotes the zero [see pdf for notation] matrix over A. Some advantage is gained in working in a general algebra. The algebra A may be taken as a Banach algebra of bounded linear transformations from a Banach space into itself and we obtain, in that instance, the results in [2] as the special case [see pdf for notation]. We may choose the elements in A to be square matrices and regard (1.1) as another partitioning A. One application of block diagonalization is to functions of matrices or, more generally, to any map f having the property: [see pdf for notation]. This would include generalized (1) - and (1,2) - inverses [6] and limits of a sequence of functions. Another application is to eigenvalues. The problem posed (1.3) leads to a natural generalization of the classical eigenvalue problem in which the eigenvalues and eigenvectors are both treated as matrices and are related by a linear equation (Section 2). By block diagonalization methods one can obtain eigenvalues and eigenvectors while simultaneously "reducing" the size of the matrix i.e., [see pdf for notation]. However, block diagonalization leads to quadratic matrix equations with matrix coefficients. In Section 5 we show that the sought-after roots can be obtained, under appropriate conditions, by means of the contraction mapping principle. These results are intended to demonstrate that solvability is possible. Better results can be gained in future work with the aid of more sophisticated fixed point theorems. In the case p = 1, and A is the complex or real numbers, the problem (1.1) reduces to the classical eigenvalue Problem for A. As a consequence of our approach we arrive at a quadratic equation whose roots are related to the eigenvectors of A. Thus we may apply fixed point theory to obtain extremal properties of positive eigenvectors (Corollary 5.1). We also obtain a realization of the eigenvalues as fixed points of a map from a subset of A into A (Section 4). A third application is to invariant subspace of A (now regarded as a transformation) since, in fact, block diagonalization and invariant subspaces are, in a sense, equivalent (Lemma 2.1, Lemma 2.2, Theorem 3.1, Theorem 3.2). Some formulas for block iagonalization and triangularization are given in Section 3.
Date: 1975-03

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