c5s2 - 5.2 Diagonalization 1 Chapter 5. Eigenvalues and...

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5.2 Diagonalization 1 Chapter 5. Eigenvalues and Eigenvectors 5.2 Diagonalization Recall. A matrix is diagonal if all entries oF the main diagonal are 0. Note. In this section, the theorems stated are valid for matrices and vectors with complex entries and complex scalars, unless stated otherwise. Theorem 5.2. Matrix Summary of Eigenvalues of A . Let A be an n × n matrix and let λ 1 2 ,...,λ n be (possibly complex) scalars and ~v 1 ,~v 2 ,...,~v n be nonzero vectors in n -space. Let C be the n × n matrix having j as j th column vector and let D = λ 1 00 ··· 0 0 λ 2 0 0 λ 3 0 . . . . . . . . . . . . . . . 000 λ n . Then AC = CD if and only if λ 1 2 n are eigenvalues of A and j is an eigenvector of A corresponding to λ j for j =1 , 2 ,...,n.
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5.2 Diagonalization 2 Proof. We have CD = . . . . . . . . . ~v 1 2 ··· n . . . . . . . . . λ 1 00 0 0 λ 2 0 0 λ 3 0 . . . . . . . . . . . . . . . 000 λ n = . . . . . . . . . λ 1 1 λ 2 2 λ n n . . . . . . . . . . Also, AC = A . . . . . . . . . 1 2 n . . . . . . . . . . Therefore, AC = if and only if A~v j = λ j j . QED Note. The n × n matrix C is invertible if and only if rank( C )= n
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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c5s2 - 5.2 Diagonalization 1 Chapter 5. Eigenvalues and...

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