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# 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 off 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 , λ 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 v j as j th column vector and let D = λ 1 0 0 · · · 0 0 λ 2 0 · · · 0 0 0 λ 3 · · · 0 . . . . . . . . . . . . . . . 0 0 0 · · · λ n . Then AC = CD if and only if λ 1 , λ 2 , . . . , λ n are eigenvalues of A and v 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 v 2 · · · v n . . . . . . . . . λ 1 0 0 · · · 0 0 λ 2 0 · · · 0 0 0 λ 3 · · · 0 . . . . . . . . . . . . . . . 0 0 0 · · · λ n = . . . . . . . . . λ 1 v 1 λ 2 v 2 · · · λ n v n . . . . . . . . . .
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