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**Unformatted text preview: **Diagonalizable matrices Theorem 1. Let A be a real n n matrix and suppose that there exists a basis of R n which consists of eigenvectors; i.e. there are n linearly independent vectors u 1 ,..., u n , and real numbers 1 , ..., n so that (1) Au i = i u i , i = 1 , . . . , n. Let P be the matrix whose columns are u 1 , ..., u n (in this order). Then P is invert- ible and (2) P- 1 AP = D := 1 . . . n so that D is the diagonal matrix with entries 1 , . . . , n on the diagonal. Vice versa if there is an invertible matrix P such that (2) holds and if u 1 , . . . , u n are the columns of P then the vectors u i form a basis of eigenvectors with AU i = i u i . Theorem 2 Let A be a complex n n matrix and suppose that there exists a basis of C n which consists of eigenvectors; i.e. there are n linearly independent vectors u 1 ,..., u n , and complex numbers 1 , ..., n so that (1) Au i = i u i , i = 1 , . . . , n. Let P be the matrix whose columns are u 1 , ..., u n (in this order). Then P is invert- ible and (2) P- 1 AP = D = 1 . . . n Vice versa if there is an invertible matrix P such that (2) holds and if u 1 , . . . , u n are the columns of P then the vectors u i form a basis of eigenvectors. The proofs of these theorems are the same. Proof: Suppose the vectors u k , k = 1 , . . . , n , form a basis of eigenvectors. With P as defined above note that P e k = u k where e 1 , . . . , e n is the usual standard basis....

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