Question

线性代数，矩阵运算

Answer 1

AP = P∧， 则 A = P∧P^(-1)
(P, E) =
[-1 1 1 1 0 0]
[ 1 0 2 0 1 0]
[ 1 1 -1 0 0 1]
初等行变换为
[ 1 0 2 0 1 0]
[ 0 1 3 1 1 0]
[ 0 1 -3 0 -1 1]
初等行变换为
[ 1 0 2 0 1 0]
[ 0 1 3 1 1 0]
[ 0 0 -6 -1 -2 1]
初等行变换为
[ 1 0 0 -1/3 1/3 1/3]
[ 0 1 0 1/2 0 1/2]
[ 0 0 1 1/6 1/3 -1/6]
P^(-1) =

[-1/3 1/3 1/3]
[1/2 0 1/2]
[1/6 1/3 -1/6]
A^n = P∧P^(-1)P∧P^(-1)P∧P^(-1) ...... P∧P^(-1)P∧P^(-1)
= P∧^nP^(-1)
φ(A) = A^3+2A^2-3A = P(∧^3 + 2∧^2-3∧)P^(-1)
= Pdiag(0, 10, 0)P^(-1) =

[5 0 5]
[0 0 0]
[5 0 5]