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M294F09WS_Eigenvalues

M294F09WS_Eigenvalues - Math 2940 Worksheet Ch 7...

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Math 2940 Worksheet: Ch. 7 Eigenvalues and Eigenvectors November 5, 2009 1. Find all eigenvalues, with their multiplicities, for the matrix 2 0 0 0 2 1 0 0 2 1 2 0 2 1 2 1 2. Find the eigenvalues and eigenvectors of the matrix 5 - 4 2 - 1 3. If A is a 2 × 2 matrix with tr A = 1 and det A = - 12, what are the eigenvalues of A ? 4. Suppose matrix A is similar to matrix B , (that is, AS = SB for some invertible matrix S , or equivalently, A = SBS - 1 ). What is the relationship between the eigenvalues of A and B ? How about the eigenvectors? 5. Suppose A is an invertible n × n matrix with 2 as an eigenvalue that has a corresponding eigenvector ~v . Is ~v an eigenvector for A - 1 ? If it is, what is the corresponding eigenvalue? 6. Let A be as above. Is ~v an eigevector for A 5 ? If so, what is the associated eigenvalue?
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7. Arguing geometrically, find all eigenvectors and eigenvalues of the linear transormations given. If possible, find a basis of R n consisting of eigenvectors.
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