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hmwk7 - eigenvalues e i and i of A satisfy the equation A...

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Department of Chemical Engineering University of California, Santa Barbara Eng 5A Fall 2001 Instructor: David Pine Homework #7 Homework due: Wednesday, 13 November 2001 1. Consider the matrix A = 1 2 1 4 3 4 0 1 2 1 - 2 3 2 3 1 1 . (1) (a) Use Mathematica to find the eigenvalues and eigenvectors of A . Hints: Make your life easier by expressing the eigenvalues { λ i } and eigenvectors { e i } in numerical (deci- mal) form. Recall that you can extract pieces of vectors and matrices in Mathematica using commands like A[[i]] and A[[i,j]] . (b) As a check on your work in part (a), find the eigenvalues of A by finding the roots of the equation det( A - λ I ) = 0. (c) Show by explicit calculation of A · e i and λ i e i that the each of the eigenvectors and
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Unformatted text preview: eigenvalues e i and λ i of A satisfy the equation A · e i = λ i e i . (2) (d) Use Mathematica to show that the four eigenvectors are linearly independent and therefore form a basis which spans R 4 . Useful point: The Chop command in Mathemat-ica replaces values smaller than 10-10 with 0 and can be useful for obtaining cleaner output when doing numerical work in Mathematica . (e) Find a 4 × 4 matrix B such that B-1 AB is diagonal. Demonstrate that B-1 AB is diagonal by explicit calculation using Mathematica . Note that the diagonal elements of B-1 AB are the eigenvalues of A ....
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