# Tagged: complex conjugate

## Problem 425

(a) Prove that each complex $n\times n$ matrix $A$ can be written as
$A=B+iC,$ where $B$ and $C$ are Hermitian matrices.

(b) Write the complex matrix
$A=\begin{bmatrix} i & 6\\ 2-i& 1+i \end{bmatrix}$ as a sum $A=B+iC$, where $B$ and $C$ are Hermitian matrices.

## 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 269

Let $A$ be a real skew-symmetric matrix, that is, $A^{\trans}=-A$.
Then prove the following statements.

(a) Each eigenvalue of the real skew-symmetric matrix $A$ is either $0$ or a purely imaginary number.

(b) The rank of $A$ is even.

## Problem 202

Show that eigenvalues of a Hermitian matrix $A$ are real numbers.

(The Ohio State University Linear Algebra Exam Problem)

## Problem 191

Let
$A=\begin{bmatrix} 1 & -1\\ 2& 3 \end{bmatrix}.$

Find the eigenvalues and the eigenvectors of the matrix
$B=A^4-3A^3+3A^2-2A+8E.$

(Nagoya University Linear Algebra Exam Problem)