Normed Vector Space

Given a complex-valued \(m \times n\) matrix \(A\), \(A^H\) denotes its hermitian conjugate. \(\ell_2^{m\times n}[Z]\) denotes the Hilbert space of sequences of \(m\times n\) complex-valued matrices, with inner product defined as

\[\langle H,G\rangle = \sum_{k=-\infty}^{\infty} \mathrm{trace}(G^H[k]H[k]).\]

The \(\ell_2^{m\times n}[Z]\) space can be decomposed as the direct sum of two spaces of sequences \(\ell_2^{m\times n}[Z^+] \oplus \ell_2^{m\times n}[Z^-]\) . The unilateral z-transform of \(G\in \ell_2^{m\times n}[Z+]\) is \(\hat{G}(z) = \sum_{k=0}^\infty G[k] z^{-k}\)

\({\mathcal{H}_2^{m \times n}}^\perp\) is the set of functions \(\hat{G} : \mathbb{C} \to \mathbb{C}^{m \times n}\) such that

\[\frac{1}{2\pi j} \oint_{\mathcal{C}} \hat{G}(z) z^{n-1} \, dz = \mathbf{0}, \quad \forall n \geq 0\]

There is a unique decomposition into \(\mathcal{H}_2\) and \(\mathcal{H}_2^\perp\) for all matrix-valued functions analytic on the unit circle, by the Projection Theorem.

\(\mathcal{RH}_2\) and \(\mathcal{RH}_2^\perp\) respectively denote the rational proper transfer function matrices in \(\mathcal{H}_2\) and \(\mathcal{H}_2^\perp\).