• I have $M$ square complex matrices $\mathbf{R}_m$ of size $N\times N$.
• I have $M$ complex, matrix-valued functions $\mathbf{H}_m(z)$ of complex variable $z$, also of size $N \times N$ .
• Each $\mathbf{H}_m(z)$ is diagonal, with all $N$ diagonal entries containing the same scalar function $H_m(z)$, i.e. $\mathbf{H}_m(z) = H_m(z)\mathbf{I}$.
• All scalar functions $H_m(z)$ are positive-real.

Is there a way to proof that $\sum_{m=1}^M \mathbf{R}_m \mathbf{H}_m(z)$ will be positive-real if all matrices $\mathbf{R}_m$ are Hermitian positive definite?

In my question I am considering $z$ as the complex variable in the $z$-transform $X(z)$ of a discrete-time sequence $x[n]$:

• I have $M$ square complex matrices $\mathbf{R}_m$ of size $N\times N$.
• I have $M$ complex, matrix-valued functions $\mathbf{H}_m(z)$ of complex variable $z$, also of size $N \times N$ .
• Each $\mathbf{H}_m(z)$ is diagonal, with all $N$ diagonal entries containing the same scalar function $H_m(z)$, i.e. $\mathbf{H}_m(z) = H_m(z)\mathbf{I}$.
• All scalar functions $H_m(z)$ are positive-real.

Is there a way to prove that $\sum_{m=1}^M \mathbf{R}_m \mathbf{H}_m(z)$ will be positive-real if all matrices $\mathbf{R}_m$ are Hermitian positive definite?

The only information that I found on discrete-time positive-real functions, is here, but it does not refer to matrix-valued functions. It says: a complex valued function of a complex variable $X(z)$ is said to be positive-real if

1. $z$ real $\implies$ $X(z)$ real
2. $|z| \geq 1 \implies \Re\{X(z)\}\geq0$