M294F09WS_Diagonalization - Math 2940 Worksheet: Ch. 7...

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Math 2940 Worksheet: Ch. 7 Eigenvalues and Eigenvectors November 12, 2009 1. Determine if the following matrix is diagonalizable. If possible, find an invertible S and a diagonal D such that S - 1 AS = D . A = 2 0 1 0 1 0 0 0 1 2. For which values of constants a , b , and c are the following matrices diagonalizable? (a) 1 a b 0 2 c 0 0 3 (b) 0 0 0 1 0 a 0 1 0
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To Diagonalize an n × n Matrix, A . . . 1. Find all eigenvalues with their algebraic multiplicities. To do this, solve the equation det( A - λI ) = 0 for λ . 2. For each eigenvalue, λ , find the corresponding eigenspace, E λ = ker( A - λI ). The dimension of E λ is the geometric multiplicity of λ . 3. If the geometric multiplicity of each eigenvalue is the same as its algebraic multiplic- ity, or equivalently, if the geometric multiplicities add up to n , then the matrix is diagonalizable. 4. If the matrix is diagonalizable, collect together the bases of each eigenspace. The
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M294F09WS_Diagonalization - Math 2940 Worksheet: Ch. 7...

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