Introduction to Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors

Definition

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

  1. A scalar $\lambda$ is called an eigenvalue of $A$ if the equation $A\mathbf{x}=\lambda\mathbf{x}$ has a nonzero solution $\mathbf{x}$.
    Such a nonzero solution $\mathbf{x}$ is called an eigenvector corresponding to the eigenvalue $\lambda$.
  2. The characteristic polynomial of $A$ is the degree $n$ polynomial $p(t)=\det(A-tI)$.
  3. If $p(t)=(t-\lambda_1)^{n_1}\cdots(t-\lambda_k)^{n_k}$ is a factorization of the characteristic polynomial of $A$, where $\lambda_1, \dots, \lambda_k$ are distinct eigenvalues of $A$, then the algebraic multiplicity of the eigenvalue $\lambda_i$ is $n_i$.
Summary

Let $A$ be an $n\times n$ matrix. Let $p(t)$ be the characteristic polynomial of $A$.

  1. The degree of $p(t)$ is $n$.
  2. $\lambda$ is an eigenvalue of $A$ if and only if $p(\lambda)=\det(A-\lambda I)=0$.
  3. $A$ has at least one eigenvalue and has at most $n$ distinct eigenvalues.
  4. $A$ has at most $n$ distinct eigenvalues.
  5. The eigenvalues of a matrix $A$ are roots of the characteristic polynomial of $A$.
  6. The eigenvalues of a triangular matrix are diagonal entries.

=solution

Problems

  1. (a) True or False. If each entry of an $n \times n$ matrix $A$ is a real number, then the eigenvalues of $A$ are all real numbers.
    (b) Find the eigenvalues of the matrix
    \[B=\begin{bmatrix}
    -2 & -1\\
    5& 2
    \end{bmatrix}.\] (The Ohio State University)

  2. Find all the eigenvalues and eigenvectors of the matrix $A=\begin{bmatrix}
    3 & -2\\
    6& -4
    \end{bmatrix}$.

  3. 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.

  4. Let
    \[A=\begin{bmatrix}
    a & -1\\
    1& 4
    \end{bmatrix}\] be a $2\times 2$ matrix, where $a$ is some real number. Suppose that the matrix $A$ has an eigenvalue $3$.

    (a) Determine the value of $a$.
    (b) Does the matrix $A$ have eigenvalues other than $3$?

  5. Determine all eigenvalues and their algebraic multiplicities of the matrix
    \[A=\begin{bmatrix}
    1 & a & 1 \\
    a &1 &a \\
    1 & a & 1
    \end{bmatrix},\] where $a$ is a real number.

  6. Suppose that $\begin{bmatrix}
    1 \\
    1
    \end{bmatrix}$ is an eigenvector of a matrix $A$ corresponding to the eigenvalue $3$ and that $\begin{bmatrix}
    2 \\
    1
    \end{bmatrix}$ is an eigenvector of $A$ corresponding to the eigenvalue $-2$. Compute $A^2\begin{bmatrix}
    4 \\
    3
    \end{bmatrix}$.
    (Stanford University)

  7. 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)

  8. Let $A$ be a $2\times 2$ real symmetric matrix. Prove that all the eigenvalues of $A$ are real numbers by considering the characteristic polynomial of $A$.
  9. Let
    \[A=\begin{bmatrix}
    a & b\\
    -b& a
    \end{bmatrix}\] be a $2\times 2$ matrix, where $a, b$ are real numbers. Suppose that $b\neq 0$. Prove that the matrix $A$ does not have real eigenvalues.

  10. Let
    \[A=\begin{bmatrix}
    3 & -12 & 4 \\
    -1 &0 &-2 \\
    -1 & 5 & -1
    \end{bmatrix}.\] Then find all eigenvalues of $A^5$. If $A$ is invertible, then find all the eigenvalues of $A^{-1}$.

  11. Let $\lambda$ be an eigenvalue of $n\times n$ matrices $A$ and $B$ corresponding to the same eigenvector $\mathbf{x}$.
    (a) Show that $2\lambda$ is an eigenvalue of $A+B$ corresponding to $\mathbf{x}$.
    (b) Show that $\lambda^2$ is an eigenvalue of $AB$ corresponding to $\mathbf{x}$.
    (The Ohio State University)

  12. Find all eigenvalues of the following $n \times n$ matrix.
    \[
    A=\begin{bmatrix}
    0 & 0 & \cdots & 0 &1 \\
    1 & 0 & \cdots & 0 & 0\\
    0 & 1 & \cdots & 0 &0\\
    \vdots & \vdots & \ddots & \ddots & \vdots \\
    0 & 0&\cdots & 1& 0 \\
    \end{bmatrix}
    \]
  13. Find all the eigenvalues of the matrix $A=\begin{bmatrix}
    0 & 1 & 0 & 0 \\
    0 &0 & 1 & 0 \\
    0 & 0 & 0 & 1 \\
    1 & 0 & 0 & 0
    \end{bmatrix}$.
    (The Ohio State University)

  14. Let $A$ be an $n \times n$ matrix. We say that $A$ is idempotent if $A^2=A$.
    (a) Find a nonzero, nonidentity idempotent matrix.
    (b) Show that eigenvalues of an idempotent matrix $A$ is either $0$ or $1$.
    (The Ohio State University)

  15. A square matrix $A$ is called idempotent if $A^2=A$.
    (a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$.
    Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$. Prove that $P$ is an idempotent matrix.
    (b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be unit vectors in $\R^n$ such that $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.
    Let $Q=\mathbf{u}\mathbf{u}^{\trans}+\mathbf{v}\mathbf{v}^{\trans}$. Prove that $Q$ is an idempotent matrix.
    (c) Prove that each nonzero vector of the form $a\mathbf{u}+b\mathbf{v}$ for some $a, b\in \R$ is an eigenvector corresponding to the eigenvalue $1$ for the matrix $Q$ in part (b).

  16. Let $A$ be an $n \times n$ matrix. Suppose that the matrix $A^2$ has a real eigenvalue $\lambda>0$. Then show that either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of the matrix $A$.
  17. Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same.

  18. For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by
    \[A=\begin{bmatrix}
    \cos\theta & -\sin\theta & 0 \\
    \sin\theta &\cos\theta &0 \\
    0 & 0 & 1
    \end{bmatrix}.\] (a) Find the determinant of the matrix $A$.
    (b) Show that $A$ is an orthogonal matrix.
    (c) Find the eigenvalues of $A$.

  19. Let $A$ be an $n\times n$ matrix. Assume that every vector $\mathbf{x}$ in $\R^n$ is an eigenvector for some eigenvalue of $A$. Prove that there exists $\lambda\in \R$ such that $A=\lambda I$, where $I$ is the $n\times n$ identity matrix.

  20. Let $A$ be a square matrix. Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$.

  21. Let $A$ be an $n$ by $n$ matrix with entries in complex numbers $\C$. Its only eigenvalues are $1,2,3,4,5$, possibly with multiplicities. What is the rank of the matrix $A+I_n$, where $I_n$ is the identity $n$ by $n$ matrix.
    (University of California, Berkeley)

  22. Let $A$ be an $n \times n$ nilpotent matrix, that is, $A^m=O$ for some positive integer $m$, where $O$ is the $n \times n$ zero matrix.
    Prove that $A$ is a singular matrix and also prove that $I-A, I+A$ are both nonsingular matrices, where $I$ is the $n\times n$ identity matrix.

  23. Determine whether each of the following statements is True or False.
    (a) If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$.
    (b) If the characteristic polynomial of an $n \times n$ matrix $A$ is $p(\lambda)=(\lambda-1)^n+2$, then $A$ is invertible.
    (c) If $A^2$ is an invertible $n\times n$ matrix, then $A^3$ is also invertible.
    (d) If $A$ is a $3\times 3$ matrix such that $\det(A)=7$, then $\det(2A^{\trans}A^{-1})=2$.

    (e) If $\mathbf{v}$ is an eigenvector of an $n \times n$ matrix $A$ with corresponding eigenvalue $\lambda_1$, and if $\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_2$, then $\mathbf{v}+\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_1+\lambda_2$.
    (Stanford University, Linear Algebra Exam Problem)

  24. Let $A=(a_{ij})$ be an $n \times n$ matrix. We say that $A=(a_{ij})$ is a right stochastic matrix if each entry $a_{ij}$ is nonnegative and the sum of the entries of each row is $1$. That is, we have
    \[a_{ij}\geq 0 \quad \text{ and } \quad a_{i1}+a_{i2}+\cdots+a_{in}=1\] for $1 \leq i, j \leq n$. Let $A=(a_{ij})$ be an $n\times n$ right stochastic matrix. Then show the following statements.
    (a)The stochastic matrix $A$ has an eigenvalue $1$.
    (b) The absolute value of any eigenvalue of the stochastic matrix $A$ is less than or equal to $1$.

  25. Suppose that $\lambda$ and $\mu$ are two distinct eigenvalues of a square matrix $A$ and let $\mathbf{x}$ and $\mathbf{y}$ be eigenvectors corresponding to $\lambda$ and $\mu$, respectively. If $a$ and $b$ are nonzero numbers, then prove that $a \mathbf{x}+b\mathbf{y}$ is not an eigenvector of $A$ (corresponding to any eigenvalue of $A$).

  26. Suppose that $A$ is an $n\times n$ singular matrix. Prove that for sufficiently small $\epsilon>0$, the matrix $A-\epsilon I$ is nonsingular, where $I$ is the $n \times n$ identity matrix.

  27. Let $A$ and $B$ be $n \times n$ matrices. Prove that the characteristic polynomials for the matrices $AB$ and $BA$ are the same.

  28. Let $A$ be an $n\times n$ real matrix. Prove that if $\lambda$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue of $A$.
  29. Let $A$ be a square matrix. Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$.
  30. Let $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities. What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix?
    (The Ohio State University)

  31. Suppose that a real symmetric matrix $A$ has two distinct eigenvalues $\alpha$ and $\beta$. Show that any eigenvector corresponding to $\alpha$ is orthogonal to any eigenvector corresponding to $\beta$.
    (Nagoya University)

  32. Show that eigenvalues of a Hermitian matrix $A$ are real numbers.

  33. Prove that the matrix
    \[A=\begin{bmatrix}
    1 & 1.00001 & 1 \\
    1.00001 &1 &1.00001 \\
    1 & 1.00001 & 1
    \end{bmatrix}\] has one positive eigenvalue and one negative eigenvalue.
    (University of California, Berkeley)

  34. Suppose that $\lambda$ and $\mu$ are two distinct eigenvalues of a square matrix $A$ and let $\mathbf{x}$ and $\mathbf{y}$ be eigenvectors corresponding to $\lambda$ and $\mu$, respectively. If $a$ and $b$ are nonzero numbers, then prove that $a \mathbf{x}+b\mathbf{y}$ is not an eigenvector of $A$ (corresponding to any eigenvalue of $A$).
  35. Consider a polynomial
    \[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\] where $a_i$ are real numbers. Define the matrix
    \[A=\begin{bmatrix}
    0 & 0 & \dots & 0 &-a_0 \\
    1 & 0 & \dots & 0 & -a_1 \\
    0 & 1 & \dots & 0 & -a_2 \\
    \vdots & & \ddots & & \vdots \\
    0 & 0 & \dots & 1 & -a_{n-1}
    \end{bmatrix}.\] Then prove that the characteristic polynomial $\det(xI-A)$ of $A$ is the polynomial $p(x)$. The matrix is called the companion matrix of the polynomial $p(x)$.

  36. Let $A$ be an $n \times n$ matrix. Suppose that all the eigenvalues $\lambda$ of $A$ are real and satisfy $\lambda <1$. Then show that the determinant $\det(I-A) >0$, where $I$ is the $n \times n$ identity matrix.

  37. Consider the $2\times 2$ matrix
    \[A=\begin{bmatrix}
    \cos \theta & -\sin \theta\\
    \sin \theta& \cos \theta \end{bmatrix},\] where $\theta$ is a real number $0\leq \theta < 2\pi$.
    (a) Find the characteristic polynomial of the matrix $A$.
    (b) Find the eigenvalues of the matrix $A$.
    (c) Determine the eigenvectors corresponding to each of the eigenvalues of $A$.

  38. Let $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues. Show that
    (a) $\det(A)=\prod_{i=1}^n \lambda_i$.
    (b) $\tr(A)=\sum_{i=1}^n \lambda_i$.

  39. (a) A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$.
    Find $\det(A)$.
    (b) A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$.
    (c) A $2\times 2$ matrix $A$ has $\det(A)=5$ and positive integer eigenvalues. What is the trace of $A$?
    (Harvard University)

  40. Let $n$ be an odd integer and let $A$ be an $n\times n$ real matrix. Prove that the matrix $A$ has at least one real eigenvalue.
  41. Let \[A=\begin{bmatrix}
    a_0 & a_1 & \dots & a_{n-2} &a_{n-1} \\
    a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2} \\
    a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3} \\
    \vdots & \vdots & \dots & \vdots & \vdots \\
    a_{2} & a_3 & \dots & a_{0} & a_{1}\\
    a_{1} & a_2 & \dots & a_{n-1} & a_{0}
    \end{bmatrix}\] be a complex $n \times n$ matrix. Such a matrix is called circulant matrix. Prove that the determinant of the circulant matrix $A$ is given by
    \[\det(A)=\prod_{k=0}^{n-1}(a_0+a_1\zeta^k+a_2 \zeta^{2k}+\cdots+a_{n-1}\zeta^{k(n-1)}),\] where $\zeta=e^{2 \pi i/n}$ is a primitive $n$-th root of unity.

  42. Let $A$ be an $n \times n$ real matrix. Prove the followings:
    (a) The matrix $AA^{\trans}$ is a symmetric matrix.
    (b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal.
    (c) The matrix $AA^{\trans}$ is non-negative definite.
    (An $n\times n$ matrix $B$ is called non-negative definite if for any $n$ dimensional vector $\mathbf{x}$, we have $\mathbf{x}^{\trans}B \mathbf{x} \geq 0$.)
    (d) All the eigenvalues of $AA^{\trans}$ is non-negative.

  43. Let $A$ be an $n\times n$ matrix. Suppose that $\mathbf{y}$ is a nonzero row vector such that $\mathbf{y}A=\mathbf{y}$. (Here a row vector means a $1\times n$ matrix.) Prove that there is a nonzero column vector $\mathbf{x}$ such that $A\mathbf{x}=\mathbf{x}$. (Here a column vector means an $n \times 1$ matrix.)
  44. (a) Let $A$ be a real orthogonal $n\times n$ matrix. Prove that the length (magnitude) of each eigenvalue of $A$ is $1$.
    (b) Let $A$ be a real orthogonal $3\times 3$ matrix and suppose that the determinant of $A$ is $1$. Then prove that $A$ has $1$ as an eigenvalue.

  45. Let $A$ and $B$ be square matrices such that they commute each other: $AB=BA$.
    Assume that $A-B$ is a nilpotent matrix. Then prove that the eigenvalues of $A$ and $B$ are the same.

  46. Let $A$ be an $n\times n$ matrix. Suppose that $A$ has real eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ with corresponding eigenvectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$. Furthermore, suppose that $|\lambda_1| > |\lambda_2| \geq \cdots \geq |\lambda_n|$. Let
    \[\mathbf{x}_0=c_1\mathbf{u}_1+c_2\mathbf{u}_2+\cdots+c_n\mathbf{u}_n\] for some real numbers $c_1, c_2, \dots, c_n$ and $c_1\neq 0$. Define $\mathbf{x}_{k+1}=A\mathbf{x}_k$ for k=0, 1, 2,\dots and let
    \[\beta_k=\frac{\mathbf{x}_k\cdot \mathbf{x}_{k+1}}{\mathbf{x}_k \cdot \mathbf{x}_k}=\frac{\mathbf{x}_k^{\trans} \mathbf{x}_{k+1}}{\mathbf{x}_k^{\trans} \mathbf{x}_k}.\] Prove that $\lim_{k\to \infty} \beta_k=\lambda_1$.