I keep seeing $$\overline{a}^H \text{ } \overline{R}_{xx} \text{ } \overline{a} = \text{single value}$$
Where $\overline{a}$ is a mx1 vector and $R_{xx}$ is a mxm autocorrelation matrix of another vector $\overline{x}$ (not same as $\overline{a}$)
But I am struggling to see what this multiple of three matrices may intuitively be doing and the purpose. Why multiply a matrix by a vector and also the hermitian transpose of that same vector. I am studying phased array antennas and direction of arrival and beamweight methods.
Many thanks,
In the context of array signal processing, the quantity $a^{H}R_{xx}a$ denotes a power as a function of the vector $a$. E.g., $a$ denotes a steering vector (with some direction-dependent underlying parameters), or more general, a filter vector. To see why this quadratic form refers to a power, let's check first the definition of the correlation matrix / covariance matrix. Assuming zero-mean signals $x$, the matrix $R$ is defined as
\begin{equation} R_{xx} = \mathcal{E}\left[xx^{H}\right] \end{equation}
with $\mathcal{E}\left[\cdot\right]$ denoting the expectation operator (assuming that x is a random variable) and $(\cdot)^H$ denotoing conjugate transpose. Hence, the quadratic reads
\begin{equation} a^{H}R_{xx}a = a^{H}\mathcal{E}\left[xx^{H}\right]a = \mathcal{E}\left[a^{H}xx^{H}a\right] = \mathcal{E}\left[|a^{H}x|^{2}\right] \geq 0 \end{equation}.