PS7 - A = ± r cos θ-r sin θ r sin θ r cos θ ² ∈ R 2 × 2 Here r> 0 Problem#5 Let A = ± 13 9-16 37 ² and B = ±-35 45 20 10 ² a Compute

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Mathematic 104, Fall 2010: Assignment #7 Due: Wednesday, December 3rd Instructions: Please ensure that your answers are legible. Also make sure that all steps are shown – even for problems consisting of a numerical answer. Bonus problems cover advanced material and, while good practice, are not required and will not be graded. Problem #1. Excercise 2.5 of Lecture 2 of Trefethen-Bau. (In part a) you may use the Schur factorization, in part c) it is helpful to think about the factorization ( I - S )( I + S ) = I - S 2 ). Problem #2. Excercise 3.2 of Lecture 3 of Trefethen-Bau. (Here || A || means the induced matrix norm on A from || · || on C m ). Problem #3. Consider the matrix: A = ± 1 2 0 - 3 0 0 1 2 0 1 ² Find the singular values of A . You do not need to compute the full SVD. (Hint: There is a right way to do this and a wrong way). Problem #4. Find the eigenvalues and eigenvectors of
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Unformatted text preview: A = ± r cos θ-r sin θ r sin θ r cos θ ² ∈ R 2 × 2 . Here r > 0. Problem #5. Let A = ± 13 9-16 37 ² and B = ±-35 45 20 10 ² . a) Compute the Schur factorization of A and of B . (Hint: Use the proof of Theorem 24.9 of Trefethen-Bau and the fact that the characteristic polynomials of A and B are easy to factor to find these factorizations). b) Determine, for both A and B , whether the matrix is diagonalizable. If it is not explain why not and if it is diagonalize it. Bonus Problem. We say a matrix A ∈ C m × m is normal if A * A = AA * . Show using the Schur factorization that if A is normal then A is unitarily diagonalizable. That is A = Q Λ Q * for Λ diagonal and Q unitary....
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This note was uploaded on 12/05/2010 for the course MATH 104 at Stanford.

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