math110s-hw9sol

# math110s-hw9sol - MATH 110 LINEAR ALGEBRA SPRING 2007/08...

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Unformatted text preview: MATH 110: LINEAR ALGEBRA SPRING 2007/08 PROBLEM SET 9 SOLUTIONS 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 .] Solution. Note that ( I- A )( I + A )- 1 = ( I + A )- 1 ( I- A ) iff ( I + A )( I- A )( I + A )- 1 = ( I- A ) iff ( I + A )( I- A ) = ( I- A )( I + A ) and this last equation is evidently true since both sides equal I- A 2 . (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. Solution. Let S := ( I + Q )- 1 ( I- Q ). Since ( A- 1 ) > = ( A > )- 1 for any nonsingular matrix A , and since Q > Q = I = QQ > , we get S > = ( I- Q ) > [( I + Q )- 1 ] > = ( I- Q > )( I + Q > )- 1 = ( QQ >- Q > )( QQ > + Q > )- 1 = [( Q- I ) Q > ][( Q + I ) Q > ]- 1 = ( Q- I ) Q > ( Q > )- 1 ( Q + I )- 1 = ( Q- I )( Q + I )- 1 =- S. So S is skew symmetric. (c) Let S ∈ R n × n be a skew symmetric matrix. Show that I- S I + S is an orthogonal matrix. Date : May 16, 2008 (Version 1.0). 1 Solution. Let Q := ( I + S )- 1 ( I- S ). Since ( A- 1 ) > = ( A > )- 1 for any nonsingular matrix A , and since S > =- S , we get, Q > = ( I- S ) > [( I + S )- 1 ] > = ( I- S > )( I + S > )- 1 = ( I + S )( I- S )- 1 = [( I- S )( I + S )- 1 ]- 1 = Q- 1 . (d) Why is it unnecessary to require that I + S be nonsingular in (c)? [Hint: Problem 3 below.] Solution. By Problem 3 (d), x > ( I + S ) x = x > I x for all x ∈ R n . Since x > I x = x > x = k x k > 0 for all x 6 = , I (and thus I + S ) is positive definite. Hence I + S is always nonsingular by Problem 3 (a). 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 ....
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math110s-hw9sol - MATH 110 LINEAR ALGEBRA SPRING 2007/08...

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