Linear Algebra, 2020a

446 Chapter Five. Similarity
1.12 Example We can use the Cayley-Hamilton Theorem to find the minimal
polynomial of this matrix.
⎛
⎞
2 0 0 1
1 2 0 2
T = ⎜
⎟
⎝0 0 2 −1⎠
0 0 0 1
First we find its characteristic polynomial c(x) =(x − 1)(x − 2) 3 with the usual
determinant |T − xI|. With that, the Cayley-Hamilton Theorem says that T’s
minimal polynomial is either (x − 1)(x − 2) or (x − 1)(x − 2) 2 or (x − 1)(x − 2) 3 .
We can decide among the choices just by computing
⎛
⎞ ⎛
⎞ ⎛
⎞
1 0 0 1 0 0 0 1 0 0 0 0
1 1 0 2
1 0 0 2
(T − 1I)(T − 2I) = ⎜
⎟ ⎜
⎟
⎝0 0 1 −1⎠
⎝0 0 0 −1⎠ = 1 0 0 1
⎜
⎟
⎝0 0 0 0⎠
0 0 0 0 0 0 0 −1 0 0 0 0
and
⎛
⎞ ⎛
⎞ ⎛
⎞
0 0 0 0 0 0 0 1 0 0 0 0
(T − 1I)(T − 2I) 2 1 0 0 1
1 0 0 2
= ⎜
⎟ ⎜
⎟
⎝0 0 0 0⎠
⎝0 0 0 −1⎠ = 0 0 0 0
⎜
⎟
⎝0 0 0 0⎠
0 0 0 0 0 0 0 −1 0 0 0 0
and so m(x) =(x − 1)(x − 2) 2 .
Exercises
̌ 1.13 What are the possible minimal polynomials if a matrix has the given characteristic
polynomial?
(a) (x − 3) 4 (b) (x + 1) 3 (x − 4) (c) (x − 2) 2 (x − 5) 2
(d) (x + 3) 2 (x − 1)(x − 2) 2
What is the degree of each possibility?
1.14 For this matrix
⎛
⎞
0 1 0 1
⎜1 0 1 0
⎟
⎝0 1 0 1⎠
1 0 1 0
find the characteristic polynomial and the minimal polynomial.
1.15 Find the minimal polynomial of this matrix.
⎛ ⎞
0 1 0
⎝0 0 1⎠
1 0 0
̌ 1.16 Find the minimal polynomial of each matrix.