Let $A$ and $B$ be $n\times n$ matrices.
Suppose that $A$ and $B$ have the same eigenvalues $\lambda_1, \dots, \lambda_n$ with the same corresponding eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$.
Prove that if the eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$ are linearly independent, then $A=B$.

Since $A$ and $B$ have $n$ linearly independent eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$, they are diagonalizable.
Specifically, if we put $S=[\mathbf{x}_1, \dots, \mathbf{x}_n]$.

Then $S$ is invertible (as column vectors of $S$ are linearly independent) and we have
\[S^{-1}AS=D \text{ and } S^{-1}BS=D,\]
where $D$ is the diagonal matrix whose diagonal entries are eigenvalues:
\[D=\begin{bmatrix}
\lambda_1 & 0 & \cdots & 0 \\
0 & \lambda_2 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \lambda_n
\end{bmatrix}.\]
It follows that we have
\[S^{-1}AS=D=S^{-1}BS,\]
and hence $A=B$. This completes the proof.

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 […]

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 […]

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 […]

True or False. Every Diagonalizable Matrix is Invertible
Is every diagonalizable matrix invertible?
Solution.
The answer is No.
Counterexample
We give a counterexample. Consider the $2\times 2$ zero matrix.
The zero matrix is a diagonal matrix, and thus it is diagonalizable.
However, the zero matrix is not […]

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 $A$ and a diagonal matrix $D$ such that […]

Basis with Respect to Which the Matrix for Linear Transformation is Diagonal
Let $P_1$ be the vector space of all real polynomials of degree $1$ or less. Consider the linear transformation $T: P_1 \to P_1$ defined by
\[T(ax+b)=(3a+b)x+a+3,\]
for any $ax+b\in P_1$.
(a) With respect to the basis $B=\{1, x\}$, find the matrix of the linear transformation […]

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 […]