math102HW8

math102HW8 - Math 102 - Homework 8 (selected problems)...

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Math 102 - Homework 8 (selected problems) David Lipshutz Problem 1. (Strang, 5.5: #14) In the list below, which classes of matrices contain A and which contain B ? A = 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 and B = 1 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Orthogonal , invertible , projection , permutation , Hermitian , rank -1, diagonalizable , Markov . Find the eigenvalues of A and B . Proof. A is orthogonal, invertible, not projection (since A 6 = A 2 ), permutation, not Hermitian (since A 6 = A T ), diagonalizable and Markov. The characteristic polynomial for A is p ( λ ) = λ 4 - 1, so the eigenvalues of A are ± 1 and ± i . B is projection (onto the space spanned by (1 , 1 , 1 , 1)), Hermitian, rank-1, diagonalizable, Markov. The eigenvalues of B are 1 since it is a Markov matrix and the rest 0 since it is rank-1. Problem 2. (Strang, 5.5: #16) Write one significant fact about the eigenvalues of each of the following. (a) A real symmetric matrix. (b) A stable matrix: all solutions to du/dt = Au approach zero. (c) An orthogonal matrix. (d) A Markov matrix. (e) A defective matrix (nondiagonalizable). (f) A singular matrix. Proof. (a) Every eigenvalue is real. (b) All eigenvalues have norm less than 1. (c) Eigenvalues all have norm equal to 1. (d) 1 is an eigenvalue of the matrix and the other eigenvalues are less than 1. (e) The matrix has repeated eigenvalues. (f) 0 is an eigenvalue of the matrix. 1
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Problem 3. (Strang, 5.5: #18) Show that a unitary matrix has | det U | = 1, but possibly det U is different from det U H . Describe all 2 by 2 matrices that are unitary. Proof. If λ 1 ,...,λ n are the eigenvalues of U , then | det U | = | λ 1 ··· λ n | = | λ 1 |···| λ n | = 1. Suppose U = " 1 0 0 i # then U H = " 1 0 0 - i # so det U = i and det U H = - i . Suppose U = " r 1 e 1 r 3 e 3 r 2 e 2 r 4 e 4 # is unitary. Then U has orthonormal columns, so r 2 1 + r 2 2 = 1 = r 2 3 + r 2 4 . Let r 1 = sin ϕ 1 then r 2 = cos ϕ 1 , and if r 3 = cos ϕ 2 then r 4 = sin ϕ 2 where 0 ϕ 1 2 π 2 . By the orthogonality of the column vectors, sin ϕ 1 cos ϕ 2 e i ( θ 1 + θ
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This note was uploaded on 04/03/2011 for the course MATH 102 taught by Professor Szypowsk during the Fall '08 term at UCSD.

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math102HW8 - Math 102 - Homework 8 (selected problems)...

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