11.8. EXERCISES 211

9. For U an orthogonal matrix, explain why ||Ux|| = ||x|| for any vector x. Next ex-plain why if U is an n×n matrix with the property that ||Ux||= ||x|| for all vectors,x, then U must be orthogonal. Thus the orthogonal matrices are exactly those whichpreserve distance. This was done in general in the chapter for unitary matrices. Doit here for the special case that the matrix is orthogonal. It will be simpler.

10. A quadratic form in three variables is an expression of the form a1x2 +a2y2 +a3z2 +a4xy+a5xz+a6yz. Show that every such quadratic form may be written as

(x y z

)A

 xyz

where A is a symmetric matrix.

11. Given a quadratic form in three variables, x,y, and z, show there exists an orthogonal

matrix U and variables x′,y′,z′ such that(

x y z)T

= U(

x′ y′ z′)T

withthe property that in terms of the new variables, the quadratic form is

λ 1(x′)2

+λ 2(y′)2

+λ 3(z′)2

where the numbers, λ 1,λ 2, and λ 3 are the eigenvalues of the matrix A in Problem10.

12. If A is a symmetric invertible matrix, is it always the case that A−1 must be symmetricalso? How about Ak for k a positive integer? Explain.

13. If A,B are symmetric matrices, does it follow that AB is also symmetric?

14. Suppose A,B are symmetric and AB = BA. Does it follow that AB is symmetric?

15. Here are some matrices. What can you say about the eigenvalues of these matricesjust by looking at them?

(a)

 0 0 00 0 −10 1 0



(b)

 1 2 −32 1 4−3 4 7



(c)

 0 −2 −32 0 −43 4 0



(d)

 1 2 30 2 30 0 2



16. Find the eigenvalues and eigenvectors of the matrix

 c 0 00 0 −b0 b 0

 . Here b,c are

real numbers.

17. Find the eigenvalues and eigenvectors of the matrix

 c 0 00 a −b0 b a

. Here a,b,c

are real numbers.