# Category: Linear Algebra

## Problem 405

Recall that a complex matrix is called Hermitian if $A^*=A$, where $A^*=\bar{A}^{\trans}$.
Prove that every Hermitian matrix $A$ can be written as the sum
$A=B+iC,$ where $B$ is a real symmetric matrix and $C$ is a real skew-symmetric matrix.

## Problem 404

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 $A$.

## Problem 403

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
$\mathbf{x}_{k+1}=A\mathbf{x}_k \text{ for } k=0, 1, 2,\dots$ 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
$\lim_{k\to \infty} \beta_k=\lambda_1.$

## Problem 400

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)

## Problem 397

Suppose $A$ is a positive definite symmetric $n\times n$ matrix.

(a) Prove that $A$ is invertible.

(b) Prove that $A^{-1}$ is symmetric.

(c) Prove that $A^{-1}$ is positive-definite.

(MIT, Linear Algebra Exam Problem)

## Problem 396

A real symmetric $n \times n$ matrix $A$ is called positive definite if
$\mathbf{x}^{\trans}A\mathbf{x}>0$ for all nonzero vectors $\mathbf{x}$ in $\R^n$.

(a) Prove that the eigenvalues of a real symmetric positive-definite matrix $A$ are all positive.

(b) Prove that if eigenvalues of a real symmetric matrix $A$ are all positive, then $A$ is positive-definite.

## Problem 395

Suppose that the vectors
$\mathbf{v}_1=\begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} -4 \\ 0 \\ -3 \\ -2 \\ 1 \end{bmatrix}$ are a basis vectors for the null space of a $4\times 5$ matrix $A$. Find a vector $\mathbf{x}$ such that
$\mathbf{x}\neq0, \quad \mathbf{x}\neq \mathbf{v}_1, \quad \mathbf{x}\neq \mathbf{v}_2,$ and
$A\mathbf{x}=\mathbf{0}.$

(Stanford University, Linear Algebra Exam Problem)

## Problem 394

Determine the values of $x$ so that the matrix
$A=\begin{bmatrix} 1 & 1 & x \\ 1 &x &x \\ x & x & x \end{bmatrix}$ is invertible.
For those values of $x$, find the inverse matrix $A^{-1}$.

## Problem 393

(a) Let $A$ be a $6\times 6$ matrix and suppose that $A$ can be written as
$A=BC,$ where $B$ is a $6\times 5$ matrix and $C$ is a $5\times 6$ matrix.

Prove that the matrix $A$ cannot be invertible.

(b) Let $A$ be a $2\times 2$ matrix and suppose that $A$ can be written as
$A=BC,$ where $B$ is a $2\times 3$ matrix and $C$ is a $3\times 2$ matrix.

Can the matrix $A$ be invertible?

## Problem 392

Let $V$ be the subspace of $\R^4$ defined by the equation
$x_1-x_2+2x_3+6x_4=0.$ Find a linear transformation $T$ from $\R^3$ to $\R^4$ such that the null space $\calN(T)=\{\mathbf{0}\}$ and the range $\calR(T)=V$. Describe $T$ by its matrix $A$.

## Problem 391

(a) Is the matrix $A=\begin{bmatrix} 1 & 2\\ 0& 3 \end{bmatrix}$ similar to the matrix $B=\begin{bmatrix} 3 & 0\\ 1& 2 \end{bmatrix}$?

(b) Is the matrix $A=\begin{bmatrix} 0 & 1\\ 5& 3 \end{bmatrix}$ similar to the matrix $B=\begin{bmatrix} 1 & 2\\ 4& 3 \end{bmatrix}$?

(c) Is the matrix $A=\begin{bmatrix} -1 & 6\\ -2& 6 \end{bmatrix}$ similar to the matrix $B=\begin{bmatrix} 3 & 0\\ 0& 2 \end{bmatrix}$?

(d) Is the matrix $A=\begin{bmatrix} -1 & 6\\ -2& 6 \end{bmatrix}$ similar to the matrix $B=\begin{bmatrix} 1 & 2\\ -1& 4 \end{bmatrix}$?

## Problem 390

Prove that if $A$ and $B$ are similar matrices, then their determinants are the same.

## Problem 389

(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, Linear Algebra Exam Problem)

## Problem 388

Let $A$ be $n\times n$ matrix and let $\lambda_1, \lambda_2, \dots, \lambda_n$ be all the eigenvalues of $A$. (Some of them may be the same.)

For each positive integer $k$, prove that $\lambda_1^k, \lambda_2^k, \dots, \lambda_n^k$ are all the eigenvalues of $A^k$.

## Problem 387

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, Linear Algebra Final Exam Problem)

## Problem 386

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.

## Problem 385

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

## Problem 384

Let $A$ be an $n\times n$ matrix with the characteristic polynomial
$p(t)=t^3(t-1)^2(t-2)^5(t+2)^4.$ Assume that the matrix $A$ is diagonalizable.

(a) Find the size of the matrix $A$.

(b) Find the dimension of the eigenspace $E_2$ corresponding to the eigenvalue $\lambda=2$.

(c) Find the nullity of $A$.

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

## Problem 383

Let
$A=\begin{bmatrix} 1 & 1 & 1 \\ 0 &0 &1 \\ 0 & 0 & 1 \end{bmatrix}$ be a $3\times 3$ matrix. Then find the formula for $A^n$ for any positive integer $n$.

## Problem 382

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, Linear Algebra Final Exam Problem)