## Common Eigenvector of Two Matrices and Determinant of Commutator

## Problem 13

Let $A$ and $B$ be $n\times n$ matrices.

Suppose that these matrices have a common eigenvector $\mathbf{x}$.

Show that $\det(AB-BA)=0$.

Read solution

Let $A$ and $B$ be $n\times n$ matrices.

Suppose that these matrices have a common eigenvector $\mathbf{x}$.

Show that $\det(AB-BA)=0$.

Read solution

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.

An $n\times n$ matrix $A$ is called **nilpotent** if $A^k=O$, where $O$ is the $n\times n$ zero matrix.

Prove the followings.

**(a)** The matrix $A$ is nilpotent if and only if all the eigenvalues of $A$ is zero.

**(b)** The matrix $A$ is nilpotent if and only if $A^n=O$.

Read solution

Let $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues.

Show that

**(1) ** $$\det(A)=\prod_{i=1}^n \lambda_i$$

**(2)** $$\tr(A)=\sum_{i=1}^n \lambda_i$$

Here $\det(A)$ is the determinant of the matrix $A$ and $\tr(A)$ is the trace of the matrix $A$.

Namely, prove that (1) the determinant of $A$ is the product of its eigenvalues, and (2) the trace of $A$ is the sum of the eigenvalues.

Read solution

Let $A= \begin{bmatrix}

1 & 2\\

2& 1

\end{bmatrix}$.

Compute $A^n$ for any $n \in \N$.

Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same.

Add to solve later