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notes5-3 - SECTION 5.3 DIAGONALIZATION EXAMPLE Use the...

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SECTION 5.3 DIAGONALIZATION EXAMPLE. Use the factorization A = PDP - 1 shown below to find a simple formula for A k , where k is an arbitrary positive integer. Then discuss what happens to A k as k → ∞ . " - 2 1 - 15 / 2 7 / 2 # = " 2 1 5 3 # " 1 / 2 0 0 1 # " 3 - 1 - 5 2 # A square matrix A is said to be diagonalizable if A is similar to a diagonal matrix, that is, if A = PDP - 1 for some invertible matrix P and some diagonal matrix D . Which matrices are diagonalizable?
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A p × p matrix A is diagonalizable exactly when A has p linearly independent eigenvectors. It turns out that the dimension of the eigenspace for an eigenvalue is less than or equal to
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