## Eigenvalues of a Matrix and its Transpose are the Same

## Problem 508

Let $A$ be a square matrix.

Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$.

Let $A$ be a square matrix.

Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$.

Prove that if $A$ is a diagonalizable nilpotent matrix, then $A$ is the zero matrix $O$.

Add to solve later Let

\[A=\begin{bmatrix}

1 & -14 & 4 \\

-1 &6 &-2 \\

-2 & 24 & -7

\end{bmatrix} \quad \text{ and }\quad \mathbf{v}=\begin{bmatrix}

4 \\

-1 \\

-7

\end{bmatrix}.\]
Find $A^{10}\mathbf{v}$.

You may use the following information without proving it.

The eigenvalues of $A$ are $-1, 0, 1$. The eigenspaces are given by

\[E_{-1}=\Span\left\{\, \begin{bmatrix}

3 \\

-1 \\

-5

\end{bmatrix} \,\right\}, \quad E_{0}=\Span\left\{\, \begin{bmatrix}

-2 \\

1 \\

4

\end{bmatrix} \,\right\}, \quad E_{1}=\Span\left\{\, \begin{bmatrix}

-4 \\

2 \\

7

\end{bmatrix} \,\right\}.\]

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

Diagonalize the matrix

\[A=\begin{bmatrix}

1 & 1 & 1 \\

1 &1 &1 \\

1 & 1 & 1

\end{bmatrix}.\]
Namely, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

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

For which values of constants $a, b$ and $c$ is the matrix

\[A=\begin{bmatrix}

7 & a & b \\

0 &2 &c \\

0 & 0 & 3

\end{bmatrix}\]
diagonalizable?

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

Determine whether the matrix

\[A=\begin{bmatrix}

1 & 4\\

2 & 3

\end{bmatrix}\]
is diagonalizable.

If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

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

Read solution

Let

\[A=\begin{bmatrix}

1 & 2 & 1 \\

-1 &4 &1 \\

2 & -4 & 0

\end{bmatrix}.\]
The matrix $A$ has an eigenvalue $2$.

Find a basis of the eigenspace $E_2$ corresponding to the eigenvalue $2$.

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

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, Linear Algebra Final Exam Problem*)

Prove that the determinant of an $n\times n$ skew-symmetric matrix is zero if $n$ is odd.

Add to solve laterLet $T:\R^2 \to \R^2$ be a linear transformation and let $A$ be the matrix representation of $T$ with respect to the standard basis of $\R^2$.

Prove that the following two statements are equivalent.

**(a)** There are exactly two distinct lines $L_1, L_2$ in $\R^2$ passing through the origin that are mapped onto themselves:

\[T(L_1)=L_1 \text{ and } T(L_2)=L_2.\]

**(b)** The matrix $A$ has two distinct nonzero real eigenvalues.

Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$.

**(a)** If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$.

**(b)** Let

\[A^{100}=aA^2+bA+cI,\]
where $I$ is the $3\times 3$ identity matrix.

Using the Cayley-Hamilton theorem, determine $a, b, c$.

(*Kyushu University, Linear Algebra Exam Problem*)

Read solution

Let $A$ be an $n\times n$ real skew-symmetric matrix.

**(a)** Prove that the matrices $I-A$ and $I+A$ are nonsingular.

**(b)** Prove that

\[B=(I-A)(I+A)^{-1}\]
is an orthogonal matrix.

Let

\[A=\begin{bmatrix}

1 & 2\\

4& 3

\end{bmatrix}.\]

**(a)** Find eigenvalues of the matrix $A$.

**(b)** Find eigenvectors for each eigenvalue of $A$.

**(c)** Diagonalize the matrix $A$. That is, find an invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

**(d)** Diagonalize the matrix $A^3-5A^2+3A+I$, where $I$ is the $2\times 2$ identity matrix.

**(e)** Calculate $A^{100}$. (You do not have to compute $5^{100}$.)

**(f)** Calculate

\[(A^3-5A^2+3A+I)^{100}.\]
Let $w=2^{100}$. Express the solution in terms of $w$.

Let

\[A=\begin{bmatrix}

1-a & a\\

-a& 1+a

\end{bmatrix}\]
be a $2\times 2$ matrix, where $a$ is a complex number.

Determine the values of $a$ such that the matrix $A$ is diagonalizable.

(*Nagoya University, Linear Algebra Exam Problem*)

Determine whether the matrix

\[A=\begin{bmatrix}

0 & 1 & 0 \\

-1 &0 &0 \\

0 & 0 & 2

\end{bmatrix}\]
is diagonalizable.

If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

Add to solve later Let $A$ be an $n\times n$ real symmetric matrix.

Prove that there exists an eigenvalue $\lambda$ of $A$ such that for any vector $\mathbf{v}\in \R^n$, we have the inequality

\[\mathbf{v}\cdot A\mathbf{v} \leq \lambda \|\mathbf{v}\|^2.\]

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*)

Read solution

Let $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.

Add to solve later Let $A$ and $B$ be $n\times n$ matrices.

Suppose that $A$ and $B$ have the same eigenvalues $\lambda_1, \dots, \lambda_n$ with the same corresponding eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$.

Prove that if the eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$ are linearly independent, then $A=B$.