# Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$

## Problem 533

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 matrix $A$ is diagonalizable.
When $A$ is diagonalizable, find a diagonal matrix $D$ so that $P^{-1}AP=D$ for some nonsingular matrix $P$.

Contents

## Solution.

Let us first find the eigenvalues of the matrix $A$.
To do so, we compute the characteristic polynomial $p(t)=\det(A-tI)$ of $A$ as follows.
Using Sarrus’s rule to compute the $3\times 3$ determinant, we have
\begin{align*}
&p(t)=\det(A-tI)\6pt] &=\begin{bmatrix} \sqrt{2}\cos x -t & i \sin x & 0 \\ i \sin x & -t &-i \sin x \\ 0 & -i \sin x & -\sqrt{2} \cos x-t \end{bmatrix}\\[6pt] &=-t(\sqrt{2}\cos x-t)(-\sqrt{2}\cos x -t) -\left(\, -(\sin^2 x) (-\sqrt{2}\cos x-t)-(\sin^2 x) (\sqrt{2}\cos x -t) \,\right)\\ &=-t^3+2(\cos^2 x-\sin ^2 x)t\\ &=-t^3+2\cos(2x) t. \end{align*} The eigenvalues of A are the roots of \[p(t)=-t^3+2\cos(2x) t=-t(t^2-2\cos(2x)). Hence the eigenvalues are
$t=0, \quad\pm \sqrt{2\cos(2x)}.$

Note that if $\sqrt{2\cos(2x)}=-\sqrt{2\cos(2x)}$ then we have $\cos(2x)=0$ and hence $x=\pi/4, 3\pi/4$.
It follows that if $x=\pi/4, 3\pi/4$, then the matrix $A$ has only one eigenvalue $0$ with algebraic multiplicity $3$.
Since $A$ is not the zero matrix, the rank of $A$ is greater than or equal to $1$.

Hence the nullity of $A$ is less than or equal to $2$ by the rank-nullity theorem.
It follows that the geometric multiplicity (=nullity) of the eigenvalue $0$ is strictly less than the algebraic multiplicity of $0$ and $A$ is not diagonalizable in this case.

Now suppose that $x\neq \pi/4, 3\pi/4$.
In this case, the matrix $A$ has three distinct eigenvalues $0, \pm \sqrt{2\cos(2x)}$.
This implies that $A$ is diagonalizable.

Let $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ be eigenvectors corresponding to eigenvalues $0, \pm \sqrt{2\cos(2x)}$, respectively.
Define the $3\times 3$ matrix $P$ by $P=\begin{bmatrix} \mathbf{v}_1 & \mathbf{v}_2 & \mathbf{v}_3 \\ \end{bmatrix}$.

It follows from the general procedure of the diagonalization that $P$ is a nonsingular matrix and
$P^{-1}AP=D,$ where $D$ is a diagonal matrix
$D=\begin{bmatrix} 0 & 0 & 0 \\ 0 &\sqrt{2\cos(2x)} &0 \\ 0 & 0 & -\sqrt{2\cos(2x)} \end{bmatrix}.$

### Summary

In summary, when $x=\pi/4, 3\pi/4$ the matrix $A$ is not diagonalizable.

When $x \neq \pi/4, 3\pi/4$, the matrix $A$ is diagonalizable and we can take the diagonal matrix $D$ as
$D=\begin{bmatrix} 0 & 0 & 0 \\ 0 &\sqrt{2\cos(2x)} &0 \\ 0 & 0 & -\sqrt{2\cos(2x)} \end{bmatrix}.$

### More from my site

• Diagonalize the 3 by 3 Matrix if it is Diagonalizable Determine whether the matrix $A=\begin{bmatrix} 0 & 1 & 0 \\ -1 &0 &0 \\ 0 & 0 & 2 \end{bmatrix}$ is diagonalizable. If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   How to […]
• Diagonalize a 2 by 2 Symmetric Matrix Diagonalize the $2\times 2$ matrix $A=\begin{bmatrix} 2 & -1\\ -1& 2 \end{bmatrix}$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   Solution. The characteristic polynomial $p(t)$ of the matrix $A$ […]
• 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 […]
• Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary Matrix Consider the Hermitian matrix $A=\begin{bmatrix} 1 & i\\ -i& 1 \end{bmatrix}.$ (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, find the eigenvectors. (c) Diagonalize the Hermitian matrix $A$ by a unitary matrix. Namely, find a diagonal matrix […]
• A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix Prove that the matrix $A=\begin{bmatrix} 0 & 1\\ -1& 0 \end{bmatrix}$ is diagonalizable. Prove, however, that $A$ cannot be diagonalized by a real nonsingular matrix. That is, there is no real nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal […]
• How to Diagonalize a Matrix. Step by Step Explanation. In this post, we explain how to diagonalize a matrix if it is diagonalizable. As an example, we solve the following problem. Diagonalize the matrix $A=\begin{bmatrix} 4 & -3 & -3 \\ 3 &-2 &-3 \\ -1 & 1 & 2 \end{bmatrix}$ by finding a nonsingular […]
• Find Eigenvalues, Eigenvectors, and Diagonalize the 2 by 2 Matrix Consider the matrix $A=\begin{bmatrix} a & -b\\ b& a \end{bmatrix}$, where $a$ and $b$ are real numbers and $b\neq 0$. (a) Find all eigenvalues of $A$. (b) For each eigenvalue of $A$, determine the eigenspace $E_{\lambda}$. (c) Diagonalize the matrix $A$ by finding a […]
• Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$ Let $A=\begin{bmatrix} 1 & 2\\ 4& 3 \end{bmatrix}.$ (a) Find eigenvalues of the matrix $A$. (b) Find eigenvectors for each eigenvalue of $A$. (c) Diagonalize the matrix $A$. That is, find an invertible matrix $S$ and a diagonal matrix $D$ such that […]

#### You may also like...

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

##### Is the Linear Transformation Between the Vector Space of 2 by 2 Matrices an Isomorphism?

Let $V$ denote the vector space of all real $2\times 2$ matrices. Suppose that the linear transformation from $V$ to...

Close