Linear Algebra

The Powers of the Matrix with Cosine and Sine Functions

Problem 567

Prove the following identity for any positive integer $n$.
\[\begin{bmatrix}
\cos \theta & -\sin \theta\\
\sin \theta& \cos \theta
\end{bmatrix}^n=\begin{bmatrix}
\cos n\theta & -\sin n\theta\\
\sin n\theta& \cos n\theta
\end{bmatrix}.\]

Add to solve later

Hint.

Recall the addition theorem of trigonometric functions (sum formula)
\begin{align*}
\sin(x+y)&=\sin x \cos y +\cos x \sin y\\
\cos(x+y)&=\cos x \cos y -\sin x \sin y.
\end{align*}

Use the induction on $n$.

Proof.

We prove the identity by induction on $n$.
The base case $n=1$ is clear.

Suppose that the identity is true for $n=k$.
Then we have
\begin{align*}
&\begin{bmatrix}
\cos \theta & -\sin \theta\\
\sin \theta & \cos \theta
\end{bmatrix}^{k+1}\\[6pt] =&
\begin{bmatrix}
\cos \theta & -\sin \theta\\
\sin \theta & \cos \theta
\end{bmatrix}
\begin{bmatrix}
\cos \theta & -\sin \theta\\
\sin \theta & \cos \theta
\end{bmatrix}^{k}\\[6pt] =&\begin{bmatrix}
\cos \theta & -\sin \theta\\
\sin \theta & \cos \theta
\end{bmatrix}
\begin{bmatrix}
\cos k\theta & -\sin k\theta\\
\sin k\theta & \cos k\theta
\end{bmatrix}
\\
&\text{(by the induction hypothesis)}\\[6pt] =&\begin{bmatrix}
\cos \theta \cos k\theta – \sin \theta \sin k \theta & -\cos \theta \sin k\theta-\sin \theta \cos k \theta\\
\sin \theta \cos k \theta + \cos \theta \sin k \theta & -\sin \theta \sin k\theta +\cos \theta \cos k \theta
\end{bmatrix}
\\
&\text{(by matrix multiplication)}\\[6pt] =&\begin{bmatrix}
\cos (\theta+k\theta) & -\sin (\theta+ k\theta)\\
\sin (\theta+k\theta) & \cos (\theta+k\theta)
\end{bmatrix}\\
&\text{(by the addition theorem of trigonometric functions)}\\[6pt] =&\begin{bmatrix}
\cos (k+1)\theta & -\sin (k+1)\theta\\
\sin (k+1)\theta & \cos (k+1)\theta
\end{bmatrix}.
\end{align*}

This proves that the identity holds for $n=k+1$.
Thus by induction, the identity is true for all positive integers $n$.

Add to solve later

More from my site

Powers of a Diagonal Matrix 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 […]
$Compute the Product $A^{2017}\mathbf{u}$ of a Matrix Power and a Vector$ 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 […]
Rotation Matrix in the Plane and its Eigenvalues and Eigenvectors Consider the $2\times 2$ matrix \[A=\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix},\] where $\theta$ is a real number $0\leq \theta < 2\pi$. (a) Find the characteristic polynomial of the matrix $A$. (b) Find the […]
Subspace Spanned By Cosine and Sine Functions Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$. Define the map $f:\R^2 \to \calF[0, 2\pi]$ by \[\left(\, f\left(\, \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta […]
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 […]
$Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$$ Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$ Consider the complex matrix \[A=\begin{bmatrix} \sqrt{2}\cos x & i \sin x & 0 \\ i \sin x &0 &-i \sin x \\ 0 & -i \sin x & -\sqrt{2} \cos x \end{bmatrix},\] where $x$ is a real number between $0$ and $2\pi$. Determine for which values of $x$ the […]
$Determine Whether Trigonometry Functions $\sin^2(x), \cos^2(x), 1$ are Linearly Independent or Dependent$ Determine Whether Trigonometry Functions $\sin^2(x), \cos^2(x), 1$ are Linearly Independent or Dependent Let $f(x)=\sin^2(x)$, $g(x)=\cos^2(x)$, and $h(x)=1$. These are vectors in $C[-1, 1]$. Determine whether the set $\{f(x), \, g(x), \, h(x)\}$ is linearly dependent or linearly independent. (The Ohio State University, Linear Algebra Midterm Exam […]
$Solve a System by the Inverse Matrix and Compute $A^{2017}\mathbf{x}$$ 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 […]