Recall that a complex matrix $M$ is said to be Hermitian if $M^*=M$.
Here $A^*$ is the conjugate transpose matrix $M^*=\bar{M}^*$.

Proof.

Let
\[B=\frac{A+A^*}{2} \text{ and } C=\frac{A-A^*}{2i}.\]
We claim that $B$ and $C$ are Hermitian matrices.
Using the fact that $(A^*)^*=A$, we compute
\begin{align*}
B^*&=\left(\, \frac{A+A^*}{2} \,\right)^*\\
&=\frac{A^*+(A^*)^*}{2}\\
&=\frac{A^*+A}{2}=B.
\end{align*}
It yields that the matrix $B$ is Hermitian.

We also have
\begin{align*}
C^*&=\left(\, \frac{A-A^*}{2i} \,\right)^*\\
&=\frac{A^*-(A^*)^*}{-2i}\\
&=\frac{A^*-A}{-2i}\\
&=\frac{A-A^*}{2i}=C.
\end{align*}
Thus, the matrix $C$ is also Hermitian.

Finally, note that we have
\begin{align*}
B+iC&=\frac{A+A^*}{2}+i\frac{A-A^*}{2i}\\
&=\frac{A+A^*}{2}+\frac{A-A^*}{2}\\
&=A.
\end{align*}
Therefore, each complex matrix $A$ can be written as $A=B+iC$, where $B$ and $C$ are Hermitian matrices.

\item By the proof of part (a), it suffices to compute
\[B=\frac{A+A^*}{2} \text{ and } C=\frac{A-A^*}{2i}.\]

We have
\[A^*=\begin{bmatrix}
-i & 2+i\\
6& 1-i
\end{bmatrix}.\]

A direct computation yields that
\[B=\begin{bmatrix}
0 & 4+\frac{i}{2}\\[6pt]
4-\frac{i}{2}& 1
\end{bmatrix} \text{ and } C=\begin{bmatrix}
1 & -\frac{1}{2}-2i\\[6pt]
-\frac{1}{2}+2i& 1
\end{bmatrix}.\]

By the result of part (a), these matrices are Hermitian and satisfy $A=B+iC$, as required.

Related Question.

Problem. 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.

Eigenvalues of a Hermitian Matrix are Real Numbers
Show that eigenvalues of a Hermitian matrix $A$ are real numbers.
(The Ohio State University Linear Algebra Exam Problem)
We give two proofs. These two proofs are essentially the same.
The second proof is a bit simpler and concise compared to the first one.
[…]

Complex Conjugates of Eigenvalues of a Real Matrix are Eigenvalues
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 $A$.
We give two proofs.
Proof 1.
Let $\mathbf{x}$ be an eigenvector corresponding to the […]

Sum of Squares of Hermitian Matrices is Zero, then Hermitian Matrices Are All Zero
Let $A_1, A_2, \dots, A_m$ be $n\times n$ Hermitian matrices. Show that if
\[A_1^2+A_2^2+\cdots+A_m^2=\calO,\]
where $\calO$ is the $n \times n$ zero matrix, then we have $A_i=\calO$ for each $i=1,2, \dots, m$.
Hint.
Recall that a complex matrix $A$ is Hermitian if […]

Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$.
Find a nonzero $3\times 3$ matrix $A$ such that $A^2\neq O$ and $A^3=O$, where $O$ is the $3\times 3$ zero matrix.
(Such a matrix is an example of a nilpotent matrix. See the comment after the solution.)
Solution.
For example, let $A$ be the following $3\times […]

Find the Eigenvalues and Eigenvectors of the Matrix $A^4-3A^3+3A^2-2A+8E$.
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)
Hint.
Apply the Cayley-Hamilton theorem.
That is if $p_A(t)$ is the […]

## 1 Response

[…] For a proof of this problem, see the post “Every complex matrix can be written as $A=B+iC$, where $B, C$ are Hermitian matrices“. […]