## If Matrices Commute $AB=BA$, then They Share a Common Eigenvector

## Problem 608

Let $A$ and $B$ be $n\times n$ matrices and assume that they commute: $AB=BA$.

Then prove that the matrices $A$ and $B$ share at least one common eigenvector.

of the day

Let $A$ and $B$ be $n\times n$ matrices and assume that they commute: $AB=BA$.

Then prove that the matrices $A$ and $B$ share at least one common eigenvector.

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 $D$ and a unitary matrix $U$ such that $U^{-1}AU=D$.

A square matrix $A$ is called **idempotent** if $A^2=A$.

Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$.

Prove that $P$ is an idempotent matrix.

Let $Q=\mathbf{u}\mathbf{u}^{\trans}+\mathbf{v}\mathbf{v}^{\trans}$.

Prove that $Q$ is an idempotent matrix.

Let

\[A=\begin{bmatrix}

1 & 2 & 1 \\

-1 &4 &1 \\

2 & -4 & 0

\end{bmatrix}.\]
The matrix $A$ has an eigenvalue $2$.

Find a basis of the eigenspace $E_2$ corresponding to the eigenvalue $2$.

(*The Ohio State University, Linear Algebra Final Exam Problem*)

Let

\[A=\begin{bmatrix}

1-a & a\\

-a& 1+a

\end{bmatrix}\]
be a $2\times 2$ matrix, where $a$ is a complex number.

Determine the values of $a$ such that the matrix $A$ is diagonalizable.

(*Nagoya University, Linear Algebra Exam Problem*)

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 $S^{-1}AS=D$.

Add to solve laterLet $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.

Add to solve later Find all the eigenvalues and eigenvectors of the matrix

\[A=\begin{bmatrix}

10001 & 3 & 5 & 7 &9 & 11 \\

1 & 10003 & 5 & 7 & 9 & 11 \\

1 & 3 & 10005 & 7 & 9 & 11 \\

1 & 3 & 5 & 10007 & 9 & 11 \\

1 &3 & 5 & 7 & 10009 & 11 \\

1 &3 & 5 & 7 & 9 & 10011

\end{bmatrix}.\]

(*MIT, Linear Algebra Homework Problem*)

Read solution

Find all eigenvalues of the matrix

\[A=\begin{bmatrix}

0 & i & i & i \\

i &0 & i & i \\

i & i & 0 & i \\

i & i & i & 0

\end{bmatrix},\]
where $i=\sqrt{-1}$. For each eigenvalue of $A$, determine its algebraic multiplicity and geometric multiplicity.

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 $S^{-1}AS=D$.

Find all the eigenvalues and eigenvectors of the matrix

\[A=\begin{bmatrix}

3 & 9 & 9 & 9 \\

9 &3 & 9 & 9 \\

9 & 9 & 3 & 9 \\

9 & 9 & 9 & 3

\end{bmatrix}.\]

(*Harvard University, Linear Algebra Final Exam Problem*)

Let $A$ be an $n \times n$ matrix and let $c$ be a complex number.

**(a)** For each eigenvalue $\lambda$ of $A$, prove that $\lambda+c$ is an eigenvalue of the matrix $A+cI$, where $I$ is the identity matrix. What can you say about the eigenvectors corresponding to $\lambda+c$?

**(b)** Prove that the algebraic multiplicity of the eigenvalue $\lambda$ of $A$ is the same as the algebraic multiplicity of the eigenvalue $\lambda+c$ of $A+cI$ are equal.

**(c)** How about geometric multiplicities?

Let $A$ be an $n\times n$ idempotent complex matrix.

Then prove that $A$ is diagonalizable.

**(a)** Let

\[A=\begin{bmatrix}

0 & 0 & 0 & 0 \\

1 &1 & 1 & 1 \\

0 & 0 & 0 & 0 \\

1 & 1 & 1 & 1

\end{bmatrix}.\]
Find the eigenvalues of the matrix $A$. Also give the algebraic multiplicity of each eigenvalue.

**(b)** Let

\[A=\begin{bmatrix}

0 & 0 & 0 & 0 \\

1 &1 & 1 & 1 \\

0 & 0 & 0 & 0 \\

1 & 1 & 1 & 1

\end{bmatrix}.\]
One of the eigenvalues of the matrix $A$ is $\lambda=0$. Find the geometric multiplicity of the eigenvalue $\lambda=0$.

Let $A, B, C$ are $2\times 2$ diagonalizable matrices.

The graphs of characteristic polynomials of $A, B, C$ are shown below. The red graph is for $A$, the blue one for $B$, and the green one for $C$.

From this information, determine the rank of the matrices $A, B,$ and $C$.

Read solution Add to solve later

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 matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

(Update 10/15/2017. A new example problem was added.)

Read solution

Let

\[ A=\begin{bmatrix}

5 & 2 & -1 \\

2 &2 &2 \\

-1 & 2 & 5

\end{bmatrix}.\]

Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix.

Your score of this problem is equal to that dimension times five.

(*The Ohio State University Linear Algebra Practice Problem*)

Read solution

Let $C$ be a $4 \times 4$ matrix with all eigenvalues $\lambda=2, -1$ and eigensapces

\[E_2=\Span\left \{\quad \begin{bmatrix}

1 \\

1 \\

1 \\

1

\end{bmatrix} \quad\right \} \text{ and } E_{-1}=\Span\left \{ \quad\begin{bmatrix}

1 \\

2 \\

1 \\

1

\end{bmatrix},\quad \begin{bmatrix}

1 \\

1 \\

1 \\

2

\end{bmatrix} \quad\right\}.\]

Calculate $C^4 \mathbf{u}$ for $\mathbf{u}=\begin{bmatrix}

6 \\

8 \\

6 \\

9

\end{bmatrix}$ if possible. Explain why if it is not possible!

(*The Ohio State University Linear Algebra Exam Problem*)

Read solution

Suppose the following information is known about a $3\times 3$ matrix $A$.

\[A\begin{bmatrix}

1 \\

2 \\

1

\end{bmatrix}=6\begin{bmatrix}

1 \\

2 \\

1

\end{bmatrix},

\quad

A\begin{bmatrix}

1 \\

-1 \\

1

\end{bmatrix}=3\begin{bmatrix}

1 \\

-1 \\

1

\end{bmatrix}, \quad

A\begin{bmatrix}

2 \\

-1 \\

0

\end{bmatrix}=3\begin{bmatrix}

1 \\

-1 \\

1

\end{bmatrix}.\]

**(a)** Find the eigenvalues of $A$.

**(b)** Find the corresponding eigenspaces.

**(c)** In each of the following questions, you must give a correct reason (based on the theory of eigenvalues and eigenvectors) to get full credit.

Is $A$ a diagonalizable matrix?

Is $A$ an invertible matrix?

Is $A$ an idempotent matrix?

(*Johns Hopkins University Linear Algebra Exam*)

Read solution

Suppose that $A$ is an $n \times n$ matrix with eigenvalue $\lambda$ and corresponding eigenvector $\mathbf{v}$.

**(a)** If $A$ is invertible, is $\mathbf{v}$ an eigenvector of $A^{-1}$? If so, what is the corresponding eigenvalue? If not, explain why not.

**(b)** Is $3\mathbf{v}$ an eigenvector of $A$? If so, what is the corresponding eigenvalue? If not, explain why not.

(*Stanford University, Linear Algebra Exam*)