- #1

RJLiberator

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## Homework Statement

Show that the sum of two nxn Hermitian matrices is Hermitian.

## Homework Equations

Hermitian conjugate means that you take the complex conjugate of the elements and transpose the matrix. I will denote it with a †.

I will denote the complex conjugate with a *.

## The Attempt at a Solution

This proof, theoretically, seems rather simple.

But, I'm just not connecting the dots. There's too much inexperience on my part.

Let me explain:

A hermitian matrix is hermitian if A†=A.

My thinking: Since we have one matrix that is hermitian, adding it to another hermitian matrix will result in a hermitian matrix. This, while not obvious, seems to make sense due to how the transpose definition works. Since we are simply adding together two hermitian matrices, the result should also be hermitian as the sum is even throughout.

If I am looking at 3x3 matrices, I note that:

[itex](

\begin{pmatrix}

a & b & c\\

d & e & f\\

g & h & i\\

\end{pmatrix}+

\begin{pmatrix}

a_1 & b_1 & c_1\\

d_1 & e_1 & f_1\\

g_1 & h_1 & i_1\\

\end{pmatrix})^†=

\begin{pmatrix}

a+a_1 & b+b_1 & c+c_1\\

d+d_1 & e+e_1 & f+f_1\\

g+g_1 & h+h_1 & i+i_1\\

\end{pmatrix}[/itex]

So I am now beginning to work with conditions.

We know the diagonal simply just has the complex conjugate to work with.

The rest can be transposed.

This is where my thinking starts to get fluttered, I feel like I went down a wrong hole. There's too much going on, it may seem. It feels like there is an easier way.