# Tagged: eigenvalue

## Problem 378

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.

## Problem 377

Let $A$ be an $n\times n$ idempotent complex matrix.
Then prove that $A$ is diagonalizable.

## Problem 376

(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 $\lambda=0$.

## Problem 374

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.
Then 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.

## Problem 373

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
$\mathbf{w}=\begin{bmatrix} 7 \\ 2 \\ -3 \end{bmatrix}.$

## Problem 363

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

(b) Let
$A=\begin{bmatrix} 1 & 0 & 3 \\ 4 &5 &6 \\ 7 & 0 & 9 \end{bmatrix} \text{ and } B=\begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 &0 \\ 0 & 0 & 4 \end{bmatrix}.$ Then find the value of
$\det(A^2B^{-1}A^{-2}B^2).$ (For part (b) without computation, you may assume that $A$ and $B$ are invertible matrices.)

## Problem 361

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 $A^{-1}$.

## Problem 357

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.

## Problem 336

A complex square ($n\times n$) matrix $A$ is called normal if
$A^* A=A A^*,$ where $A^*$ denotes the conjugate transpose of $A$, that is $A^*=\bar{A}^{\trans}$.
A matrix $A$ is said to be nilpotent if there exists a positive integer $k$ such that $A^k$ is the zero matrix.

(a) Prove that if $A$ is both normal and nilpotent, then $A$ is the zero matrix.
You may use the fact that every normal matrix is diagonalizable.

(b) Give a proof of (a) without referring to eigenvalues and diagonalization.

(c) Let $A, B$ be $n\times n$ complex matrices. Prove that if $A$ is normal and $B$ is nilpotent such that $A+B=I$, then $A=I$, where $I$ is the $n\times n$ identity matrix.

## Problem 323

Suppose that $A$ is $2\times 2$ matrix that has eigenvalues $-1$ and $3$.
Then for each positive integer $n$ find $a_n$ and $b_n$ such that
$A^{n+1}=a_nA+b_nI,$ where $I$ is the $2\times 2$ identity matrix.

## Problem 321

Let $V$ be a real vector space of all real sequences
$(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).$ Let $U$ be a subspace of $V$ defined by
$U=\{(a_i)_{i=1}^{\infty}\in V \mid a_{n+2}=2a_{n+1}+3a_{n} \text{ for } n=1, 2,\dots \}.$ Let $T$ be the linear transformation from $U$ to $U$ defined by
$T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots).$

(a) Find the eigenvalues and eigenvectors of the linear transformation $T$.

(b) Use the result of (a), find a sequence $(a_i)_{i=1}^{\infty}$ satisfying $a_1=2, a_2=7$.

## Problem 315

Let $P_1$ be the vector space of all real polynomials of degree $1$ or less. Consider the linear transformation $T: P_1 \to P_1$ defined by
$T(ax+b)=(3a+b)x+a+3,$ for any $ax+b\in P_1$.

(a) With respect to the basis $B=\{1, x\}$, find the matrix of the linear transformation $T$.

(b) Find a basis $B’$ of the vector space $P_1$ such that the matrix of $T$ with respect to $B’$ is a diagonal matrix.

(c) Express $f(x)=5x+3$ as a linear combination of basis vectors of $B’$.

## Problem 314

Let $T$ be the linear transformation from the vector space $\R^2$ to $\R^2$ itself given by
$T\left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right)= \begin{bmatrix} 3x_1+x_2 \\ x_1+3x_2 \end{bmatrix}.$

(a) Verify that the vectors
$\mathbf{v}_1=\begin{bmatrix} 1 \\ -1 \end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ are eigenvectors of the linear transformation $T$, and conclude that $B=\{\mathbf{v}_1, \mathbf{v}_2\}$ is a basis of $\R^2$ consisting of eigenvectors.

(b) Find the matrix of $T$ with respect to the basis $B=\{\mathbf{v}_1, \mathbf{v}_2\}$.

## Problem 310

Let $V$ be a real vector space of all real sequences
$(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).$ Let $U$ be the subspace of $V$ consisting of all real sequences that satisfy the linear recurrence relation
$a_{k+2}-5a_{k+1}+3a_{k}=0$ for $k=1, 2, \dots$.
Let $T$ be the linear transformation from $U$ to $U$ defined by
$T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots).$

Let $B=\{\mathbf{u}_1, \mathbf{u}_2\}$ be a basis of $U$, where
\begin{align*}
\mathbf{u}_1&=(1, 0, -3, -15, -66, \dots)\\
\mathbf{u}_2&=(0, 1, 5, 22, 95, \dots).
\end{align*}
Let $A$ be the matrix representation of the linear transformation $T: U \to U$ with respect to the basis $B$.

(a) Find the eigenvalues and eigenvectors of $T$.

(b) Use the result of (a), find a sequence $(a_i)_{i=1}^{\infty}$ satisfying the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ and the initial condition $a_1=1, a_2=1$.

(c) Find the formula for the sequences $(a_i)_{i=1}^{\infty}$ satisfying the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ and express it using $a_1, a_2$.

## Problem 269

Let $A$ be a real skew-symmetric matrix, that is, $A^{\trans}=-A$.
Then prove the following statements.

(a) Each eigenvalue of the real skew-symmetric matrix $A$ is either $0$ or a purely imaginary number.

(b) The rank of $A$ is even.

## Problem 259

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 $3$?

## Problem 258

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 $A$).

## Problem 237

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.

## Problem 235

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, Linear Algebra Final Exam Problem)

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 $A$.