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 give 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 $C$.
    Graphs of characteristic polynomials
  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.
    (c) How about geometric multiplicities?

  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)