# Tagged: symmetric matrix

## Problem 200

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 Linear Algebra Practice Problem)

## Problem 143

Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.

(a) The set $S$ consisting of all $n\times n$ symmetric matrices.

(b) The set $T$ consisting of all $n \times n$ skew-symmetric matrices.

(c) The set $U$ consisting of all $n\times n$ nonsingular matrices.

## Problem 111

Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings.

(a) The product $AB$ is symmetric if and only if $AB=BA$.

(b) If the product $AB$ is a diagonal matrix, then $AB=BA$.

## Problem 63

Let $A$ be an $n\times n$ real symmetric matrix whose eigenvalues are all non-negative real numbers.

Show that there is an $n \times n$ real matrix $B$ such that $B^2=A$.

## Problem 59

Answer the following two questions with justification.

(a) Does there exist a $2 \times 2$ matrix $A$ with $A^3=O$ but $A^2 \neq O$? Here $O$ denotes the $2 \times 2$ zero matrix.

(b) Does there exist a $3 \times 3$ real matrix $B$ such that $B^2=A$ where
$A=\begin{bmatrix} 1 & -1 & 0 \\ -1 &2 &-1 \\ 0 & -1 & 1 \end{bmatrix}\,\,\,\,?$

(Princeton University Linear Algebra Exam)

## Problem 42

Let $A$ be a $4\times 4$ real symmetric matrix. Suppose that $\mathbf{v}_1=\begin{bmatrix} -1 \\ 2 \\ 0 \\ -1 \end{bmatrix}$ is an eigenvector corresponding to the eigenvalue $1$ of $A$.
Suppose that the eigenspace for the eigenvalue $2$ is $3$-dimensional.

(a) Find an orthonormal basis for the eigenspace of the eigenvalue $2$ of $A$.

(b) Find $A\mathbf{v}$, where
$\mathbf{v}=\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}.$

(The University of Tokyo Linear Algebra Exam)

## Problem 19

Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix.

(a) Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of $B$.

(b) Show that $\tr(AA^{\trans})$ is the sum of the square of the entries of $A$.

(c) Show that if $A$ is nonzero symmetric matrix, then $\tr(A^2)>0$.

## Problem 12

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.