- Lecture 32 Andrei Antonenko 1 Powers of diagonalizable matrices In this section we will give 2 algorithms of computing the m-th power of a

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Lecture 32 Andrei Antonenko April 30, 2003 1 Powers of diagonalizable matrices In this section we will give 2 algorithms of computing the m -th power of a matrix. 1.1 Method 1 First method is based on diagonalization. Suppose A is a given matrix, and we want to find its m -th power, i.e. we want to get a formula for A m . We will suppose that the matrix A is diagonalizable. Let λ 1 2 ,...,λ n be the eigenvalues of A , and e 1 ,e 2 ,...,e n be its linearly independent eigenvectors. Then we know, that there exists a matrix C , whose columns are eigenvectors, and a diagonal matrix D = λ 1 0 ... 0 0 λ 2 ... 0 . . . . . . . . . . . . . . . 0 0 ... λ n such that D = C - 1 AC, or A = CDC - 1 . Now we can see that A m = ( CDC - 1 ) m = ( CDC - 1 )( CDC - 1 ) ··· ( CDC - 1 ) | {z } m = CD ( C - 1 C ) D ( C - 1 ...C ) DC - 1 = CDID . ..IDC - 1 = CD m C - 1 . But D m = λ 1 0 ... 0 0 λ 2 ... 0 . . . . . . . . . . . . . . . 0 0 ... λ n m = λ m 1 0 ... 0 0 λ m 2 ... 0 . . . . . . . . . . . . . . . . . 0 0 ... λ m n 1
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So, A m = C λ m 1 0 ... 0 0 λ m 2 ... 0 . . . . . . . . . . . . . . . . . 0 0 ... λ m n C - 1 . Example 1.1. Let’s find the formula for ˆ 3 - 2 1 0 ! m
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This note was uploaded on 05/28/2011 for the course AMS 2010 taught by Professor Andant during the Spring '03 term at SUNY Stony Brook.

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- Lecture 32 Andrei Antonenko 1 Powers of diagonalizable matrices In this section we will give 2 algorithms of computing the m-th power of a

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