Classroom Tips and Techniques: Simultaneous Diagonalization and the Generalized Eigenvalue Problem - Maple Application Center
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## Classroom Tips and Techniques: Simultaneous Diagonalization and the Generalized Eigenvalue Problem

Author
: Dr. Robert Lopez
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This article explores the connections between the generalized eigenvalue problem and the problem of simultaneously diagonalizing a pair of n × n matrices.

Given the n × n matrices A and B, the generalized eigenvalue problem seeks the eigenpairs (lambdak, xk), solutions of the equation Ax = lambda Bx, or (A - lambda B) x = 0. If B is nonsingular, the eigenpairs of B-1 A are solutions. If a matrix S exists for which ST A S = Lambda, and ST B S = I, where Lambda is a diagonal matrix and I is the n × n identity, then A and B are said to be diagonalized simultaneously, in which case the diagonal entries of Lambda are the generalized eigenvalues for A and B. Such a matrix S exists if A is symmetric and B is positive definite. (Our definition of positive definite includes symmetry.)

#### Application Details

Publish Date: December 06, 2011
Created In: Maple 15
Language: English

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