# Eigenvectors and Eigenspaces

## Eigenvectors and Eigenspaces

Definition

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

1. The eigenspace corresponding to an eigenvalue $\lambda$ of $A$ is defined to be $E_{\lambda}=\{\mathbf{x}\in \C^n \mid A\mathbf{x}=\lambda \mathbf{x}\}$.
Summary

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

1. The eigenspace $E_{\lambda}$ consists of all eigenvectors corresponding to $\lambda$ and the zero vector.
2. $A$ is singular if and only if $0$ is an eigenvalue of $A$.
3. The nullity of $A$ is the geometric multiplicity of $\lambda=0$ if $\lambda=0$ is an eigenvalue.

=solution

### Problems

1. Let $A$ be a $3\times 3$ matrix. Suppose that $A$ has eigenvalues $2$ and $-1$, and suppose that $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors corresponding to $2$ and $-1$, respectively, where
$\mathbf{u}=\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} \text{ and } \mathbf{v}=\begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}.$ Then compute $A^5\mathbf{w}$, where $\mathbf{w}=\begin{bmatrix} 7 \\ 2 \\ -3 \end{bmatrix}$.

2. 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)

3. 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 $\lambda=0$.

4. Let $A$ and $B$ be $n\times n$ matrices. Suppose that these matrices have a common eigenvector $\mathbf{x}$. Show that $\det(AB-BA)=0$.

5. Suppose that $A$ is a diagonalizable matrix with characteristic polynomial
$f_A(\lambda)=\lambda^2(\lambda-3)(\lambda+2)^3(\lambda-4)^3.$ (a) Find the size of the matrix $A$.
(b) Find the dimension of $E_4$, the eigenspace corresponding to the eigenvalue $\lambda=4$.
(c) Find the dimension of the nullspace of $A$.
(Stanford University)

6. Let $A$ be a square matrix and its characteristic polynomial is given by
$p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).$ Find the rank of $A$.
(The Ohio State University)

7. (a) Let
$A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21}& a_{22} \end{bmatrix}$ be a matrix such that $a_{11}+a_{12}=1$ and $a_{21}+a_{22}=1$. Namely, the sum of the entries in each row is $1$. (Such a matrix is called (right) stochastic matrix.) Then prove that the matrix $A$ has an eigenvalue $1$.
(b) Find all the eigenvalues of the matrix
$B=\begin{bmatrix} 0.3 & 0.7\\ 0.6& 0.4 \end{bmatrix}.$ (c) For each eigenvalue of $B$, find the corresponding eigenvectors.

8. Let $A$ be an $n\times n$ matrix. Suppose that $\lambda_1, \lambda_2$ are distinct eigenvalues of the matrix $A$ and let $\mathbf{v}_1, \mathbf{v}_2$ be eigenvectors corresponding to $\lambda_1, \lambda_2$, respectively. Show that the vectors $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent.

9. Let $A$ and $B$ be an $n \times n$ matrices. Suppose that all the eigenvalues of $A$ are distinct and the matrices $A$ and $B$ commute, that is $AB=BA$. Then prove that each eigenvector of $A$ is an eigenvector of $B$.

10. Let $H$ and $E$ be $n \times n$ matrices satisfying the relation $HE-EH=2E$. Let $\lambda$ be an eigenvalue of the matrix $H$ such that the real part of $\lambda$ is the largest among the eigenvalues of $H$. Let $\mathbf{x}$ be an eigenvector corresponding to $\lambda$. Then prove that $E\mathbf{x}=\mathbf{0}$.

11. 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)

12. Let $A=\begin{bmatrix} 1 & -14 & 4 \\ -1 &6 &-2 \\ -2 & 24 & -7 \end{bmatrix}$ and $\quad \mathbf{v}=\begin{bmatrix} 4 \\ -1 \\ -7 \end{bmatrix}$. Find $A^{10}\mathbf{v}$. You may use the following information without proving it. The eigenvalues of $A$ are $-1, 0, 1$. The eigenspaces are given by
$E_{-1}=\Span\left\{\, \begin{bmatrix} 3 \\ -1 \\ -5 \end{bmatrix} \,\right\}, \quad E_{0}=\Span\left\{\, \begin{bmatrix} -2 \\ 1 \\ 4 \end{bmatrix} \,\right\}, \quad E_{1}=\Span\left\{\, \begin{bmatrix} -4 \\ 2 \\ 7 \end{bmatrix} \,\right\}.$ (The Ohio State University)

13. 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 $C$. 14. 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)

15. 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.

16. 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)

17. Find the determinant of the following matrix
$A=\begin{bmatrix} 6 & 2 & 2 & 2 &2 \\ 2 & 6 & 2 & 2 & 2 \\ 2 & 2 & 6 & 2 & 2 \\ 2 & 2 & 2 & 6 & 2 \\ 2 & 2 & 2 & 2 & 6 \end{bmatrix}.$ (Harvard University)

18. 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.

19. 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)

20. Consider the matrix
$A=\begin{bmatrix} 3/2 & 2\\ -1& -3/2 \end{bmatrix} \in M_{2\times 2}(\R).$ (a) Find the eigenvalues and corresponding eigenvectors of $A$.
(b) Show that for $\mathbf{v}=\begin{bmatrix} 1 \\ 0 \end{bmatrix}\in \R^2$, we can choose $n$ large enough so that the length $\|A^n\mathbf{v}\|$ is as small as we like.
(University of California, Berkeley)

21. Let $F$ and $H$ be an $n\times n$ matrices satisfying the relation $HF-FH=-2F$.
(a) Find the trace of the matrix $F$.
(b) Let $\lambda$ be an eigenvalue of $H$ and let $\mathbf{v}$ be an eigenvector corresponding to $\lambda$. Show that there exists an positive integer $N$ such that $F^N\mathbf{v}=\mathbf{0}$.

22. 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.
23. Let $a$ and $b$ be two distinct positive real numbers. Define matrices
$A:=\begin{bmatrix} 0 & a\\ a & 0 \end{bmatrix}, \,\, B:=\begin{bmatrix} 0 & b\\ b& 0 \end{bmatrix}.$ Find all the pairs $(\lambda, X)$, where $\lambda$ is a real number and $X$ is a non-zero real matrix satisfying the relation
$AX+XB=\lambda X.$ (The University of Tokyo)