426 CHAPTER 19. EIGENVALUES AND EIGENVECTORS
2. In the above problem, suppose vk =uk. Show Tv = v if v ∈V ≡ span(u1, · · · ,ur) .Now show that T (T (x)) = T (x) .
3. The Cayley Hamilton theorem states that every matrix satisfies its characteristicequation. I have given a short proof of this major theorem in the appendix on thetheory of determinants. See Section 20.2.10. Suppose you have p(λ ) is the charac-teristic polynomial for a square n×n matrix A. Show that this matrix is invertible ifand only if the constant term of the p(λ ) is non zero. In this case, give a formula forA−1 in terms of powers of A. Say
p(λ ) = λn +an−1λ
n−1 + · · ·+a1λ +a0
Thus you need explain why a0 ̸= 0 if A−1 exists and then find a formula for A−1
when this is the case. Hint: By the Cayley Hamilton theorem p(A) = 0 meaning
An +an−1An−1 + · · ·+a1A+a0I = 0
Now consider solving for I and factoring out A.
4. Here are some matrices. Label according to whether they are symmetric, skew sym-metric, or orthogonal. If the matrix is orthogonal, determine whether it is proper orimproper.
(a)
1 0 00 1/
√2 −1/
√2
0 1/√
2 1/√
2
(b)
1 2 −32 1 4−3 4 7
(c)
0 −2 −32 0 −43 4 0
5. Show that every real matrix may be written as the sum of a skew symmetric and a
symmetric matrix. Hint: If A is an n× n matrix, show that B ≡ 12
(A−AT
)is skew
symmetric.
6. Let x be a vector in Rn and consider the matrix I − 2xxT
|x|2. Show this matrix is both
symmetric and orthogonal.
7. For U an orthogonal matrix, explain why |Ux|= |x| for any vector x. Next explainwhy if U is an n×n matrix with the property that |Ux|= |x| for all vectors, x, then Umust be orthogonal. Thus the orthogonal matrices are exactly those which preservedistance. This was done in general in the chapter for orthogonal matrices. Try to doit in your own words.
8. 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.