solhw1

# solhw1 - Solutions for Homework 1 Numerical Linear Algebra...

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Unformatted text preview: Solutions for Homework 1 Numerical Linear Algebra 1 Fall 2010 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. Solution: There are multiple ways to prove this. The simplest is to use contradiction on a matrix vector product identity. We have that a matrix A ∈ R n × n is nilpotent of degree k if k is a positive integer such that A p = 0 p ≥ k A p 6 = 0 < p < k Suppose λ 6 = 0 is an eigenvalue corresponding to the eigenvector x 6 = 0 n . It follows that Ax = λx A k x = λ k x However, by the nilpotent assumption A k = 0 and therefore A k x = 0 n × n x = 0 n = λ k x Since x 6 = 0 n it follows that λ = 0 which is a contradiction. Therefore all λ must be 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. Solution: Since A is unitary we have AA H = A H A = I and therefore A is normal. It follows that A = Q Λ Q H where Λ is a diagonal matrix with possibly complex scalars on the diagonal and Q is a unitary matrix. We therefore have I = A H A = ( Q Λ Q H ) H ( Q Λ Q H ) = Q ¯ Λ Q H Q Λ Q H = Q | Λ | 2 Q H where | Λ | 2 is a diagonal matrix with elements | λ i | 2 on the diagonal. Q | Λ | 2 Q H = I → | Λ | 2 = Q H IQ = I → | λ i | 2 = 1 So | λ i | = 1 for 1 ≤ i ≤ n as desired. 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. Solution: ← Assume A is such that U H AU = Λ is a diagonal matrix and U is unitary. We have AA H = U Λ U H ( U Λ U H ) H = U Λ U H ( U Λ U H ) H = U Λ U H U ¯ Λ U H = U Λ ¯ Λ U H = U ¯ ΛΛ U H = U ¯ Λ U H U Λ U H = ( U Λ U H ) H ( U Λ U H ) = A H A → Assume that AA H = A H A . By the Schur decomposition we have A = QUQ H where U is upper triangular and Q is unitary. It follows trivially that AA H = QUU H Q H A H A = QU H UQ H ↓ UU H = U H U So U is upper triangular and normal . It can be shown that if U is upper triangular...
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solhw1 - Solutions for Homework 1 Numerical Linear Algebra...

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