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I am trying to write a product of matrices in index notation.

Recall the product of two matrices $X,Y$ is just

$$ (XY)_{a i} = X_{ab}Y_{bi} $$ where repeated indices are summed over, in this case, $b$. My question is for the following matrix multiplication: $$ X^\top \bar{X} X^\top +X^\top X X^H $$ where $\bar{X}$ is the complex conjugate of the matrix $X$; $X^\top, X^H$ are the real transpose and hermitian transpose respectively. How can I write this in index notation?

My solution is below, if you could tell me if it is wrong or not. Thanks! $$ (X^\top \bar{X} X^\top +X^\top X X^H)_{ai} $$ The first term is: $$ (X^\top \bar{X} X^\top)_{ai}= X^\top_{ak} \bar{X}_{kb} X^\top_{b i}= X_{ka}\bar{X}_{kb}X_{ib} $$ and the second term I get : $$ (X^\top X X^H)_{ai}= X^\top_{ak} X_{kb} X^H_{bi}= X_{ka}X_{kb} \bar{X}_{ib} $$ since $X^H= \bar{X}^\top$. Thanks for your help!

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    Just for clarification, in your solution you also sum them over the same indices right? – 2017-01-19
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    As I stated, 'repeated indices are summed over'. Is this what you mean? I can write the sums out explicitly if that will help, thanks a lot! In my solution, I sum over $k,b$ for both terms. – 2017-01-19

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Using your notation

\begin{align} (X^T\bar{X}X^T)_{ai}&=(X^T)_{ak}(\bar{X}X^T)_{ki} \\ &=X_{ka}\bar{X}_{kb}X^T_{bi} \\ &=X_{ka}\bar{X}_{kb}X_{ib} \end{align}

\begin{align} (X^TXX^H)_{ai}&=(X^T)_{ak}(XX^H)_{ki} \\ &=X_{ka}X_{kb}X^H_{bi} \\ &=X_{ka}X_{kb}\bar{X}_{ib} \end{align}

Summing them up

$$X_{ka}\bar{X}_{kb}X_{ib}+X_{ka}X_{kb}\bar{X}_{ib}=2X_{ka}\Re(\bar{X}_{kb}X_{ib})$$

Here, $\Re$ refers to the real part of a complex number.