Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such that
\[S^{-1}BS=A.\]

A matrix $A$ is diagonalizable if $A$ is similar to a diagonal matrix. Namely, $A$ is diagonalizable if there exist a nonsingular matrix $S$ and a diagonal matrix $D$ such that
\[S^{-1}AS=D.\]

Some useful facts are

If $S$ and $T$ are invertible matrices, then we have
\[(TS)^{-1}=S^{-1}T^{-1}.\]
(Note the order of the product.)

A matrix is nonsingular if and only if its determinant is nonzero.

Proof.

Since the matrix $A$ is diagonalizable, there exist a nonsingular matrix $S$ and a diagonal matrix $D$ such that
\[S^{-1}AS=D. \tag{*}\]
Also, since $B$ is similar to $A$, there exist a nonsingular matrix $T$ such that
\[T^{-1}BT=A. \tag{**}\]

Inserting the expression of $A$ from (**) into the equality (*), we obtain
\begin{align*}
D&=S^{-1}(T^{-1}BT)S\\
&=(S^{-1}T^{-1})B(TS)\\
&=(TS)^{-1}B(TS) \tag{***}.
\end{align*}

Now let us put $U:=TS$. Then the matrix $U$ is nonsingular.
(This is because we have
\[\det(U)=\det(TS)=\det(T)\det(S)\neq 0\]
since $T$ and $S$ are nonsingular matrices, hence their determinants are not zero.)

Therefore from (***) we have
\[D=U^{-1}BU,\]
where $D$ is a diagonal matrix and $U$ is a nonsingular matrix.
Thus $B$ is a diagonalizable matrix.

Determine Whether Given Matrices are Similar
(a) Is the matrix $A=\begin{bmatrix}
1 & 2\\
0& 3
\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
3 & 0\\
1& 2
\end{bmatrix}$?
(b) Is the matrix $A=\begin{bmatrix}
0 & 1\\
5& 3
\end{bmatrix}$ similar to the matrix […]

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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 […]

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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 […]

Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix
Let $A$ be an $n\times n$ matrix with real number entries.
Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix.
Proof.
Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$.
The orthogonality of the […]

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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 […]

If Two Matrices are Similar, then their Determinants are the Same
Prove that if $A$ and $B$ are similar matrices, then their determinants are the same.
Proof.
Suppose that $A$ and $B$ are similar. Then there exists a nonsingular matrix $S$ such that
\[S^{-1}AS=B\]
by definition.
Then we […]

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 […]

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[…] For a proof, see the post “A matrix similar to a diagonalizable matrix is also diagonalizable“. […]

[…] For a proof, see the post “A matrix similar to a diagonalizable matrix is also diagonalizable“. […]