# Powers of a Diagonal Matrix

## Problem 7

Let $A=\begin{bmatrix} a & 0\\ 0& b \end{bmatrix}$.
Show that

(1) $A^n=\begin{bmatrix} a^n & 0\\ 0& b^n \end{bmatrix}$ for any $n \in \N$.

(2) Let $B=S^{-1}AS$, where $S$ be an invertible $2 \times 2$ matrix.
Show that $B^n=S^{-1}A^n S$ for any $n \in \N$

Contents

## Hint.

Use mathematical induction.

## Proof.

(1) We prove $A^n=\begin{bmatrix} a^n & 0\\ 0& b^n \end{bmatrix}$ by induction on $n$.
The base case $n=1$ is true by definition.

Suppose that $A^k=\begin{bmatrix} a^k & 0\\ 0& b^k \end{bmatrix}$. Then we have
$A^{k+1}=AA^k =\begin{bmatrix} a & 0\\ 0& b \end{bmatrix} \begin{bmatrix} a^k & 0\\ 0& b^k \end{bmatrix} =\begin{bmatrix} a^{k+1} & 0\\ 0& b^{k+1} \end{bmatrix}.$ Here we used the induction hypothesis in the second equality.
Hence the inductive step holds. This completes the proof.

(2) We show that $B^n=S^{-1}A^n S$ by induction on $n$.
When $n=1$, this is just the definition of $B$.

For induction step, assume that $B^k=S^{-1} A^k S$.
Then we have
$B^{k+1}=B B^k=(S^{-1} A S) (S^{-1} A^k S)=S^{-1}A A^k S=S^{-1} A^{k+1} S,$ where we used the induction hypothesis in the second equality and the third equality follows by canceling $S S^{-1}=I_2$ in the middle.

Thus the inductive step holds, and this competes the proof.

### More from my site

• The Powers of the Matrix with Cosine and Sine Functions Prove the following identity for any positive integer $n$. $\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix}^n=\begin{bmatrix} \cos n\theta & -\sin n\theta\\ \sin n\theta& \cos […] • Compute the Product A^{2017}\mathbf{u} of a Matrix Power and a Vector Let \[A=\begin{bmatrix} -1 & 2 \\ 0 & -1 \end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix} 1\\ 0 \end{bmatrix}.$ Compute $A^{2017}\mathbf{u}$.   (The Ohio State University, Linear Algebra Exam) Solution. We first compute $A\mathbf{u}$. We […]
• Problems and Solutions About Similar Matrices Let $A, B$, and $C$ be $n \times n$ matrices and $I$ be the $n\times n$ identity matrix. Prove the following statements. (a) If $A$ is similar to $B$, then $B$ is similar to $A$. (b) $A$ is similar to itself. (c) If $A$ is similar to $B$ and $B$ […]
• A Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.   Definitions/Hint. Recall the relevant definitions. Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such […]
• Dimension of Null Spaces of Similar Matrices are the Same Suppose that $n\times n$ matrices $A$ and $B$ are similar. Then show that the nullity of $A$ is equal to the nullity of $B$. In other words, the dimension of the null space (kernel) $\calN(A)$ of $A$ is the same as the dimension of the null space $\calN(B)$ of […]
• Find the Formula for the Power of a Matrix Let $A=\begin{bmatrix} 1 & 1 & 1 \\ 0 &0 &1 \\ 0 & 0 & 1 \end{bmatrix}$ be a $3\times 3$ matrix. Then find the formula for $A^n$ for any positive integer $n$.   Proof. We first compute several powers of $A$ and guess the general formula. We […]
• Solve a System by the Inverse Matrix and Compute $A^{2017}\mathbf{x}$ Let $A$ be the coefficient matrix of the system of linear equations \begin{align*} -x_1-2x_2&=1\\ 2x_1+3x_2&=-1. \end{align*} (a) Solve the system by finding the inverse matrix $A^{-1}$. (b) Let $\mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ be the solution […]
• Companion Matrix for a Polynomial Consider a polynomial $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,$ where $a_i$ are real numbers. Define the matrix \[A=\begin{bmatrix} 0 & 0 & \dots & 0 &-a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & […]

### 1 Response

1. 07/21/2016

[…] We diagonalize the matrix $A$ and use this Problem. […]

This site uses Akismet to reduce spam. Learn how your comment data is processed.

##### A Linear Transformation Maps the Zero Vector to the Zero Vector

Let $T : \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation. Let $\mathbf{0}_n$ and $\mathbf{0}_m$ be zero vectors of $\mathbb{R}^n$ and...

Close