C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes5-3

# C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes5-3 -...

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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. " 17 - 6 45 - 16 # = " 2 1 5 3 # " 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 eigenvector is less than or equal to
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Unformatted text preview: the multiplicity of that eigenvector, and consequently A is diagonizable exactly when the dimension of each eigenspace is equal to the multiplicity of the eigenvalue. Finally, when this happens, we get an eigenvector basis for R p by listing bases for each eigenspace. EXAMPLES. Diagonalize the following matrices, if possible. 1. " 3 0 2 3 # 2. 4 0-2 2 5 4 0 0 5 HOMEWORK: SECTION 5.3...
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C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes5-3 -...

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