Linear Algebra with Applications (3rd Edition)

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Math 417 homework 10 Solutions Problem 1 (a) Show: if v 1 , v 2 are eigenvectors of a matrix A for the same eigenvalue λ , then their linear combinations α 1 v 1 + α 2 v 2 are also eigenvectors (except for those = 0) for that eigenvalue λ . Answer: A ( α 1 v 1 + α 2 v 2 ) = α 1 Av 1 + α 2 Av 2 = λα 1 v 1 + λα 2 v 2 = λ ( α 1 v 1 + α 2 v 2 ) . (b) Consider A = SDS - 1 where S = 1 0 2 0 - 1 0 2 0 0 , D = - 1 0 0 0 2 0 0 0 - 1 . (1) What are the eigenvalues of A ? What are the eigenvectors? [There is a hard way and an easy way to do this.] Answer: the eigenvalues are the diagonal entries of D ; - 1 is a double root and 2 is single. The eigenvectors are the columns of S . Column i is an eigenvector for the eigenvalue in column i row i of D . (2) Find an orthogonal diagonalization of A , i.e. an orthogonal 3 × 3 matrix U with A = UDU T . Answer: we already have a diagonalization, but S is not an orthogonal ma- trix. But part (a) tells us that the Gram-Schmidt algorithm can be used to
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