A First Course in Linear Algebra, 2017a

572 Selected Exercise Answers
7.2.2 The eigenvectors and eigenvalues are:
⎧⎡
⎤⎫
⎧⎡
⎨ 2 ⎬ ⎨
⎣ 0 ⎦
⎩ ⎭ ↔ 1, ⎣
⎩
1
−2
1
0
⎤⎫
⎧⎡
⎬ ⎨
⎦
⎭ ↔ 1, ⎣
⎩
7
−2
2
⎤⎫
⎬
⎦
⎭ ↔ 3
The matrix P needed to diagonalize the above matrix is
⎡
2 −2
⎤
7
⎣ 0 1 −2 ⎦
1 0 2
and the diagonal matrix D is
⎡
⎣
1 0 0
0 1 0
0 0 3
⎤
⎦
7.2.3 The eigenvectors and eigenvalues are:
⎧⎡
⎤⎫
⎧⎡
⎨ −6 ⎬ ⎨
⎣ −1 ⎦
⎩ ⎭ ↔ 6, ⎣
⎩
−2
−5
−2
2
⎤⎫
⎧⎡
⎬ ⎨
⎦
⎭ ↔−3, ⎣
⎩
The matrix P needed to diagonalize the above matrix is
⎡
−6 −5
⎤
−8
⎣ −1 −2 −2 ⎦
2 2 3
−8
−2
3
⎤⎫
⎬
⎦
⎭ ↔−2
and the diagonal matrix D is
⎡
⎣
6 0 0
0 −3 0
0 0 −2
⎤
⎦
7.2.8 The eigenvalues are distinct because they are the n th roots of 1. Hence if X is a given vector with
X =
n
∑ a j V j
j=1
then
so A nm = I.
A nm X = A nm
∑
n a j V j =
j=1
n
∑ a j A nm V j =
j=1
n
∑ a j V j = X
j=1
7.2.13 AX =(a + ib)X. Now take conjugates of both sides. Since A is real,
AX =(a − ib)X