math110s-hw9

math110s-hw9 - MATH 110: LINEAR ALGEBRA SPRING 2007/08...

Info iconThis preview shows pages 1–2. 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: MATH 110: LINEAR ALGEBRA SPRING 2007/08 PROBLEM SET 9 1. A matrix S ∈ R n × n is called skew symmetric if S > =- S . (a) For any matrix A ∈ R n × n for which I + A is nonsingular, show that ( I- A )( I + A )- 1 = ( I + A )- 1 ( I- A ) . (1.1) We will write I- A I + A for the matrix in (1.1). [Note: In general, AB- 1 6 = B- 1 A and so A B is ambiguous since it could mean either AB- 1 or B- 1 A .] (b) Let Q ∈ R n × n be an orthorgonal matrix such that I + Q is nonsingular. Show that I- Q I + Q is a skew symmetric matrix. (c) Let S ∈ R n × n be a skew symmetric matrix. Show that I- S I + S is an orthogonal matrix. (d) Why is it unnecessary to require that I + S be nonsingular in (c)? [Hint: Problem 3 below.] 2. Let A,B ∈ R n × n . Let λ a ∈ R be an eigenvalue of A and λ b ∈ R be an eigenvalue of B . (a) Is it always true that λ a λ b is an eigenvalue of AB ? Is it always true that λ a + λ b is an eigenvalue of A + B ?...
View Full Document

This note was uploaded on 08/01/2008 for the course MATH 110 taught by Professor Gurevitch during the Spring '08 term at Berkeley.

Page1 / 2

math110s-hw9 - MATH 110: LINEAR ALGEBRA SPRING 2007/08...

This preview shows document pages 1 - 2. Sign up to view the full document.

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