# hw1 - B k = 0 for some k ≤ n Suppose A ∈ C n × n(not...

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Homework 1 Numerical Linear Algebra 1 Fall 2010 Due date: beginning of class Friday, 9/10/10 Problem 1.1 A matrix A C n × n is nilpotent if A k = 0 for some integer k > 0. Prove that the only eigenvalue of a nilpotent matrix is 0. Problem 1.2 Let the matrix A C n × n be unitary. Show that if λ is an eigenvalue of A then | λ | = 1. Problem 1.3 Prove that a matrix A C n × n is normal if and only if there exists a unitary matrix U such that U H AU is a diagonal matrix. Problem 1.4 If A C n × n then the trace of A is the sum of the diagonal elements, i.e., trace( A ) = n X i =1 e H i Ae i 1.4.a . Show that trace( AB ) = trace( BA ) where A C n × n and B C n × n . 1.4.b . Show that trace( Q H AQ ) = trace( A ) where A C n × n and Q C n × n is a unitary matrix. 1.4.c . Show that trace( A ) = n i =1 λ i where λ i are the eigenvalues of A . Problem 1.5 Recall that a nilpotent n × n matrix B is such that

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Unformatted text preview: B k = 0 for some k ≤ n . Suppose A ∈ C n × n (not necessarily Hermitian) and show that A = A D + A N where A D is a nondefective, i.e., diagonalizable, matrix, A N is a nilpotent matrix and A D A N = A N A D . 1 Problem 1.6 Let A ∈ C n × n and deﬁne H A = 1 2 ( A + A H ) and S A = 1 2 ( A-A H ) called the Hermitian part and skew Hermitian parts of A respectively. Clearly, A = H A + S A . Show that A is a normal matrix if and only if the matrix product of H A and S A commutes, i.e., H A S A = S A H A . Problem 1.7 Problem 7.1.1 Golub and Van Loan p. 318 Problem 1.8 Problem 7.1.5 Golub and Van Loan p. 318 Problem 1.9 Problem 7.1.8 Golub and Van Loan p. 318 2...
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hw1 - B k = 0 for some k ≤ n Suppose A ∈ C n × n(not...

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