solhw1

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Solutions for Homework 1 Numerical Linear Algebra 1 Fall 2011 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 6 = 0 is an eigenvalue corresponding to the eigenvector x 6 = 0 n . It follows that Ax = x A k x = k x However, by the nilpotent assumption A k = 0 and therefore A k x = 0 n n x = 0 n = k x Since x 6 = 0 n it follows that = 0 which is a contradiction. Therefore all must be 0. Problem 1.2 Problem 7.1.1 Golub and Van Loan p. 318 Solution: Lemma: If U C n n is upper triangular and normal then it is a diagonal matrix. Proof: Partition U as follows U = U n- 1 u n- 1 H n- 1 nn where U n- 1 [ C ] n- 1 n- 1 is upper triangular and u n- 1 [ C ] n- 1 . Since U is normal we have UU H = U H U e H n UU H e n = e H n U H Ue n | nn | 2 = u H n- 1 u n- 1 + | nn | 2 1 Therefore, u n- 1 = 0 n- 1 and we have U = U n- 1 n- 1 H n- 1 nn Given this structure and since U is normal, it follows that U n- 1 is a normal upper triangular matrix. Therefore, the last column of U n- 1 also has 0 in all elements above the diagonal and we have U = U n- 2 D 2 where D 2 is a 2 2 diagonal matrix and U n- 2 is an upper triangular n- 2 n- 2 matrix. Once again, since U is normal this structure implies U n- 2 is normal. This repeats until all columns above the diagonal are shown to be 0, i.e., U is diagonal, as desired. 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....
View Full Document

Page1 / 7

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online