Calculating all potencies of a Matrix

Question

I've stumbled across this problem while reading my textbook (chapter eigenvalues)

Calculate all potencies of $A$ and $A+aE$

$ a \in K$ and $A \in K-Vectorspace$

$ A= \begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix} \quad$ and hence $\quad A+aE = \begin{pmatrix} a & 1 & 0 & 0 & 0 \\ 0 & a & 1 & 0 & 0 \\ 0 & 0 & a & 1 & 0 \\ 0 & 0 & 0 & a & 1 \\ 0 & 0 & 0 & 0 & a \end{pmatrix}\qquad $

It can't be $ A^{n} = A \cdots A$ but what are potencies of $A$ and $A+aE$, if this question popped up during the chapter Eigenvalues?

Answer 1

You aren't talking of powers, are you?

In that case you have $$ A= \begin{bmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}, \quad A^{2} = \begin{bmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}, \\ A^{3} = \begin{bmatrix} 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}, \quad A^{4} = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}, $$ and $A^{k} = 0$ for $k \ge 5$.

Then since $E A = A E$, you have \begin{align} (a E + A)^{n} &= \sum_{k=0}^{n} \binom{n}{k} (a E)^{n-k} A^{k} \\&= a^{n} E + n a^{n-1} A + \binom{n}{2} a^{n-2} A^{2} + \binom{n}{3} a^{n-3} A^{3} + \binom{n}{4} a^{n-4} A^{4} \\&= \begin{bmatrix} a^{n} & n a^{n-1} & \binom{n}{2} a^{n-2} & \binom{n}{3} a^{n-3} & \binom{n}{4} a^{n-4} \\ 0 & a^{n} & n a^{n-1} & \binom{n}{2} a^{n-2} & \binom{n}{3} a^{n-3}\\ 0 & 0 & a^{n} & n a^{n-1} & \binom{n}{2} a^{n-2}\\ 0 & 0 & 0 & a^{n} & n a^{n-1}\\ 0 & 0 & 0 & 0 & a^{n} \end{bmatrix}. \end{align}

Answer 2

The potency of a non-negative matrix $A$ is the smallest $n>0$ such that $diag(A^n) > 0$.

So for $A$, there is not any potency; for $A+aE$, if $a>0$, the potency is $1$, and if $a<0$, the potency is $2$.