Question

1. Find the Eigen values of

0 1 0
A= 0 0 1
216k^3 -108k^2 18k

lambda 1 = ?
lambda 2= ?
lambda 3= ?

2. Using the fact that the matrix

0 0 0 ... 0 -c subscript 0
1 0 0 ... 0 -c subscript 1
0 1 0 ... 0 -c subscript 2
... ... ... ... ... ...
0 0 0 ... 1 -c subscript n-1

has the characteristic polynomial

p(λ) = c0 + c1λ + ⋯ + cn − 1λn − 1 + λn

find a matrix with the characteristic polynomial

p(λ) = 1 − 5λ + λ2 + 6λ3 + λ4

Answer 1

1.

$∣ А-λ E ∣= \begin{vmatrix} -λ & 1&0 \\ 0&-λ& 1\\ 216k^3&-108k^2&18k-λ \end{vmatrix}=0\\$

$λ^2 (18к-λ )+2 1 6 к^ 3 +0-0-0-1 0 8 к^2 λ=0$

$-λ ^3 +1 8 k λ^2-1 0 8 к^2 λ+2 1 6 к^3 =0$

$λ ^3 -1 8 k λ^2 +1 0 8 к ^2 λ-2 1 6 к ^3 =0$

$( λ ^3 -2 1 6 к ^3 )-( 1 8 k λ ^2 -1 0 8 к ^2 λ)=0$

$( λ-6 к ) ( λ ^2 +6 k λ+3 6 к ^2 )-1 8 k λ ( λ-6 к )=0$

$( λ-6 к ) ( λ ^2 -1 2 k λ+3 6 к ^2 )=0$

$λ-6 к=0 ,( λ-6 к )^2 =0$

$λ _1 =6 к ,\\ λ _2 =6 к ,\\ λ _3 =6 к$

2.

the required matrix with the characteristic polynomial

$p ( λ )=1-5 λ+λ ^2 +6 λ ^3 +λ ^4$

in our case

$\begin{pmatrix} 0 &0&0&-c_0 \\ 1&0&0&-c_1\\ 0&1&0&-c_2\\ 0&0&1&-c_3 \end{pmatrix}$

in our case

$c _0 ​=1,c _1 =−5,c _2 =1,c _3 =6$

there

$\begin{pmatrix} 0 &0&0&-1 \\ 1&0&0&5\\ 0&1&0&-1\\ 0&0&1&-6 \end{pmatrix}$