# 5.2 - MATH1850U/2050U Chapter 5 cont 1 EIGENVALUES AND...

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MATH1850U/2050U: Chapter 5 cont… 1 EIGENVALUES AND EIGENVECTORS cont… Diagonalization (Section 5.2; pg. 305) Two Equivalent Problems we consider: 1. The Eigenvector Problem: Given an n n matrix A , does there exist a basis for n R consisting of eigenvectors of A ? 2. The Diagonalization Problem (Matrix Form): Given an n n matrix A , does there exist an invertible matrix P such that AP P 1 is a diagonal matrix? Definition: A square matrix A is called diagonalizable if there is an invertible matrix P such that AP P 1 is a diagonal matrix; the matrix P is said to diagonalize A . Example: Consider A and P below. Show that A is diagonalizable.

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MATH1850U/2050U: Chapter 5 cont… 2 Theorem: If A is an n n matrix, then the following are equivalent: a) A is diagonalizable. b) A has n linearly independent eigenvectors. Remark: Since A is n n , the eigenvectors are in R n . Thus, b) implies that the eigenvectors form a basis for R n . Example:
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5.2 - MATH1850U/2050U Chapter 5 cont 1 EIGENVALUES AND...

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