Where can I read about this 'rule'? I was trying to solve an equation, I got nowhere, and the solution used a 'rule' that I have never seen before, $a^TX\,b = b^TX\,a$. 
What is this rule?
Where can I read about it?
Note: $a$ and $b$ are vectors and $X$ is a matrix.
 A: I guess, $X$ is symmetric here, because in general, the following holds:
$$a^TXb= b^TX^Ta$$
which is a simple application of $(AB)^T=B^TA^T$ and the fact that the scalars as $1\times 1$ matrices are all 'symmetric': $\lambda=\lambda^T$.
More specifically:
$$a^TXb = (a^TXb)^T=b^TX^Ta$$
A: What you're seeing here is the fact that $a^TXb$ is simply just a complex number. I think your problem is missing some context, but usually an inner product $\langle \cdot, \cdot \rangle$ is defined by a symmetric positive definite matrix (let's call it $X$).  In this case we have that $\langle a, b \rangle = a^TXb$.  Noting however that the inner product takes two vectors and gives us a number, and the fact that the inner product is symmetric tells us that $\langle a, b \rangle = \langle b, a \rangle$ and therefore $a^TXb = b^TXa$. 
A: a^t is for transpose, right?
$
        \begin{pmatrix}
        a_1 & a_2  \\
        \end{pmatrix}
$$
        \begin{pmatrix}
        x_{11} & x_{12}  \\
        x_{21} & x_{22} \\
        \end{pmatrix}
$$
        \begin{pmatrix}
        b_1 \\ b_2  \\
        \end{pmatrix}
$ = $a_1b_1x_{11} + a_2b_1x_{21} + a_1b_2x_{12} + a_2b_2x_{22} $
but
$
        \begin{pmatrix}
        b_1 & b_2  \\
        \end{pmatrix}
$$
        \begin{pmatrix}
        x_{11} & x_{12}  \\
        x_{21} & x_{22} \\
        \end{pmatrix}
$$
        \begin{pmatrix}
        a_1 \\ a_2  \\
        \end{pmatrix}
$ = $a_1b_1x_{11} + a_1b_2x_{21} + a_2b_1x_{12} + a_2b_2x_{22} $
So, if X is symmetrical, ($x_{12} = x_{21}$), then this rule is true just by doing multiplication. 
