## There is at Least One Real Eigenvalue of an Odd Real Matrix

## Problem 407

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.

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.

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.)

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.

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 $A$.

Add to solve later 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

\[\lim_{k\to \infty} \beta_k=\lambda_1.\]

Let $G$ be a group. Suppose that we have

\[(ab)^3=a^3b^3\]
for any elements $a, b$ in $G$. Also suppose that $G$ has no elements of order $3$.

Then prove that $G$ is an abelian group.

Add to solve later Let $G$ be a group. Suppose that

\[(ab)^2=a^2b^2\]
for any elements $a, b$ in $G$. Prove that $G$ is an abelian group.

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*)

Read solution

Prove that the cubic polynomial $x^3-2$ is irreducible over the field $\Q(i)$.

Add to solve laterSuppose $A$ 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*)

Read solution

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.

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*)

Read solution

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 $A^{-1}$.

**(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?

Add to solve later 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 $A$.

**(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}$?

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

Add to solve later**(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*)

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 $A^k$.

Add to solve later