Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$.
The orthogonality of the matrix $Q$ means that we have
\[Q^{\trans}Q=QQ^{\trans}=I, \tag{*}\]
where $Q^{\trans}$ is the transpose matrix of $Q$ and $I$ is the $n\times n$ identity matrix.

Since $Q$ diagonalizes the matrix $A$, we have
\[Q^{-1}AQ=D,\]
where $D$ is a diagonal matrix.
Equivalently, we have
\[A=QDQ^{-1} \tag{**}.\]
Taking transpose of both sides, we obtain
\begin{align*}
A^{\trans}&=(QDQ^{-1})^{\trans}\\
&=(Q^{-1})^{\trans}D^{\trans} Q^{-1}\\
&=(Q^{-1})^{\trans}D Q^{-1} \text{ since } D \text{ is diagonal.}\tag{***}
\end{align*}

By (*), we observe that the inverse matrix of $Q$ is the transpose $Q^{\trans}$, that is, $Q^{-1}=Q^{\trans}$.
It follows from this observation and (***) that we have
\[A^{\trans}=QDQ^{-1}.\]
(Note that $(Q^{-1})^{\trans}=Q^{\trans \trans}=Q$.)

Comparing this with (**), we obtain
\[A^{\trans}=A,\]
and hence $A$ is a symmetric matrix.

Two Matrices with the Same Characteristic Polynomial. Diagonalize if Possible.
Let
\[A=\begin{bmatrix}
1 & 3 & 3 \\
-3 &-5 &-3 \\
3 & 3 & 1
\end{bmatrix} \text{ and } B=\begin{bmatrix}
2 & 4 & 3 \\
-4 &-6 &-3 \\
3 & 3 & 1
\end{bmatrix}.\]
For this problem, you may use the fact that both matrices have the same characteristic […]

Quiz 13 (Part 1) Diagonalize a Matrix
Let
\[A=\begin{bmatrix}
2 & -1 & -1 \\
-1 &2 &-1 \\
-1 & -1 & 2
\end{bmatrix}.\]
Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$.
That is, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that […]

A Square Root Matrix of a Symmetric Matrix with Non-Negative Eigenvalues
Let $A$ be an $n\times n$ real symmetric matrix whose eigenvalues are all non-negative real numbers.
Show that there is an $n \times n$ real matrix $B$ such that $B^2=A$.
Hint.
Use the fact that a real symmetric matrix is diagonalizable by a real orthogonal matrix.
[…]

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 the Inverse Matrix of a Matrix With Fractions
Find the inverse matrix of the matrix
\[A=\begin{bmatrix}
\frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\[6 pt]
\frac{6}{7} &\frac{2}{7} &-\frac{3}{7} \\[6pt]
-\frac{3}{7} & \frac{6}{7} & -\frac{2}{7}
\end{bmatrix}.\]
Hint.
You may use the augmented matrix […]

A Square Root Matrix of a Symmetric Matrix
Answer the following two questions with justification.
(a) Does there exist a $2 \times 2$ matrix $A$ with $A^3=O$ but $A^2 \neq O$? Here $O$ denotes the $2 \times 2$ zero matrix.
(b) Does there exist a $3 \times 3$ real matrix $B$ such that $B^2=A$ […]

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 […]

Construction of a Symmetric Matrix whose Inverse Matrix is Itself
Let $\mathbf{v}$ be a nonzero vector in $\R^n$.
Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$.
Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by
\[A=I-a\mathbf{v}\mathbf{v}^{\trans},\]
where […]