Realization of tensor-product and of tensor-Factorization of rational   functions

Daniel Alpay; Izchak Lewkowicz

arXiv:1812.01437·math.OC·December 5, 2018

Realization of tensor-product and of tensor-Factorization of rational functions

Daniel Alpay, Izchak Lewkowicz

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TL;DR

This paper explores the realization of tensor-products and tensor-factorizations of rational functions, providing explicit formulas and methods to handle their state space representations, especially under specific limiting conditions.

Contribution

It introduces explicit formulas for tensor-factorization of rational functions and extends classical realization methods to tensor-product cases with dimension inflation.

Findings

01

Explicit realization formulas for tensor-products of rational functions.

02

A method for tensor-factorization under certain limit conditions.

03

Extension of classical realization theory to tensor-product scenarios.

Abstract

We here first study the state space realization of a tensor-product of a pair of rational functions. At the expense of "inflating" the dimensions, we recover the classical expressions for realization of a regular product of rational functions. Then, under an additional assumption that the limit at infinity of a given rational function exists and is equal to identity, an explicit formula for a tensor-factorization of this function, is introduced.

Equations106

d_{1}^{*} M_{1} c_{1} = k = 1 \sum N_{2} (d_{1} \otimes f_{k})^{*} M (c_{1} \otimes f_{k}), c_{1}, d_{1} \in C^{N_{1}},

d_{1}^{*} M_{1} c_{1} = k = 1 \sum N_{2} (d_{1} \otimes f_{k})^{*} M (c_{1} \otimes f_{k}), c_{1}, d_{1} \in C^{N_{1}},

d_{2}^{*} M_{2} c_{2} = k = 1 \sum N_{1} (e_{k} \otimes d_{2})^{*} M (e_{k} \otimes c_{2}), c_{2}, d_{2} \in C^{N_{2}},

d_{2}^{*} M_{2} c_{2} = k = 1 \sum N_{1} (e_{k} \otimes d_{2})^{*} M (e_{k} \otimes c_{2}), c_{2}, d_{2} \in C^{N_{2}},

d_{1}^{*} R_{A} (z) c_{1} d_{2}^{*} R_{B} (z) c_{2} = k = 1 \sum N_{2} (d_{1} \otimes f_{k})^{*} R (z) (c_{1} \otimes f_{k}), = k = 1 \sum N_{2} (d_{2} \otimes f_{k})^{*} R (z) (c_{2} \otimes f_{k}),

d_{1}^{*} R_{A} (z) c_{1} d_{2}^{*} R_{B} (z) c_{2} = k = 1 \sum N_{2} (d_{1} \otimes f_{k})^{*} R (z) (c_{1} \otimes f_{k}), = k = 1 \sum N_{2} (d_{2} \otimes f_{k})^{*} R (z) (c_{2} \otimes f_{k}),

R_{A} (z) R_{B} (z) = R_{1} (z) \cdot (Tr R_{2} (z)) = R_{2} (z) \cdot (Tr R_{1} (z))

R_{A} (z) R_{B} (z) = R_{1} (z) \cdot (Tr R_{2} (z)) = R_{2} (z) \cdot (Tr R_{1} (z))

\begin{matrix}F_{l}(z_{l})=D_{l}+C_{l}(z_{l}I_{n_{l}}-A_{l})^{-1}B_{l}&&&F_{r}(z_{r})=D_{r}+C_{r}(z_{r}I_{n_{r}}-A_{r})^{-1}B_{r}\\ \leavevmode\nobreak\ \\ R_{F_{l}}=\footnotesize\left(\begin{array}[]{c|c}A_{l}&B_{l}\\ \hline\cr C_{l}&D_{l}\end{array}\right)&&&R_{F_{r}}=\footnotesize\left(\begin{array}[]{c|c}A_{r}&B_{r}\\ \hline\cr C_{r}&D_{r}\end{array}\right).\end{matrix}

\begin{matrix}F_{l}(z_{l})=D_{l}+C_{l}(z_{l}I_{n_{l}}-A_{l})^{-1}B_{l}&&&F_{r}(z_{r})=D_{r}+C_{r}(z_{r}I_{n_{r}}-A_{r})^{-1}B_{r}\\ \leavevmode\nobreak\ \\ R_{F_{l}}=\footnotesize\left(\begin{array}[]{c|c}A_{l}&B_{l}\\ \hline\cr C_{l}&D_{l}\end{array}\right)&&&R_{F_{r}}=\footnotesize\left(\begin{array}[]{c|c}A_{r}&B_{r}\\ \hline\cr C_{r}&D_{r}\end{array}\right).\end{matrix}

\begin{matrix}R_{F_{l}F_{r}}=\footnotesize\left(\begin{array}[]{cc|c}A_{l}&B_{l}C_{r}&B_{l}D_{r}\\ 0&A_{r}&B_{r}\\ \hline\cr C_{l}&D_{l}C_{r}&D_{l}D_{r}\end{array}\right)=\footnotesize\left(\begin{array}[]{c|c}A&B\\ \hline\cr C&D\end{array}\right)=\left(\begin{smallmatrix}A_{l}&0&B_{l}\\ 0&I_{n_{r}}&0\\ C_{l}&0&D_{l}\end{smallmatrix}\right)\left(\begin{smallmatrix}I_{n_{l}}&0&0\\ 0&A_{r}&B_{r}\\ 0&C_{r}&D_{r}\end{smallmatrix}\right),\end{matrix}

\begin{matrix}R_{F_{l}F_{r}}=\footnotesize\left(\begin{array}[]{cc|c}A_{l}&B_{l}C_{r}&B_{l}D_{r}\\ 0&A_{r}&B_{r}\\ \hline\cr C_{l}&D_{l}C_{r}&D_{l}D_{r}\end{array}\right)=\footnotesize\left(\begin{array}[]{c|c}A&B\\ \hline\cr C&D\end{array}\right)=\left(\begin{smallmatrix}A_{l}&0&B_{l}\\ 0&I_{n_{r}}&0\\ C_{l}&0&D_{l}\end{smallmatrix}\right)\left(\begin{smallmatrix}I_{n_{l}}&0&0\\ 0&A_{r}&B_{r}\\ 0&C_{r}&D_{r}\end{smallmatrix}\right),\end{matrix}

F_{1} (z_{1}) F_{2} (z_{2}) = D_{1} D_{2} + (C_{1} D_{1} C_{2}) ((z_{1} I_{n_{1}} 0 0 z_{2} I_{n_{2}}) - (A_{1} 0 0 A_{2}))^{- 1} (B_{1} D_{2} B_{2}) .

F_{1} (z_{1}) F_{2} (z_{2}) = D_{1} D_{2} + (C_{1} D_{1} C_{2}) ((z_{1} I_{n_{1}} 0 0 z_{2} I_{n_{2}}) - (A_{1} 0 0 A_{2}))^{- 1} (B_{1} D_{2} B_{2}) .

n_{l} + n_{r} \leavevmode .

n_{l} + n_{r} \leavevmode .

A_{l} := A_{l} \otimes I_{p_{r}} B_{l} := B_{l} \otimes I_{p_{r}} C_{l} := C_{l} \otimes I_{p_{r}} D_{l} := D_{l} \otimes I_{p_{r}} F_{l} (z_{l}) := C_{l} (z_{l} I_{n_{l} p_{r}} - A_{l})^{- 1} B_{l} + D_{l} A_{r} := I_{m_{l}} \otimes A_{r} B_{r} := I_{m_{l}} \otimes B_{r} C_{r} := I_{m_{l}} \otimes C_{r} D_{r} := I_{m_{l}} \otimes D_{r} F_{r} (z_{r}) := C_{r} (z_{l} I_{m_{l} n_{r}} - A_{r})^{- 1} B_{r} + D_{r} \leavevmode .

A_{l} := A_{l} \otimes I_{p_{r}} B_{l} := B_{l} \otimes I_{p_{r}} C_{l} := C_{l} \otimes I_{p_{r}} D_{l} := D_{l} \otimes I_{p_{r}} F_{l} (z_{l}) := C_{l} (z_{l} I_{n_{l} p_{r}} - A_{l})^{- 1} B_{l} + D_{l} A_{r} := I_{m_{l}} \otimes A_{r} B_{r} := I_{m_{l}} \otimes B_{r} C_{r} := I_{m_{l}} \otimes C_{r} D_{r} := I_{m_{l}} \otimes D_{r} F_{r} (z_{r}) := C_{r} (z_{l} I_{m_{l} n_{r}} - A_{r})^{- 1} B_{r} + D_{r} \leavevmode .

R_{F_{l} \otimes F_{r}} = R_{F_{l} F_{r}} \leavevmode .

R_{F_{l} \otimes F_{r}} = R_{F_{l} F_{r}} \leavevmode .

T X \otimes Y Z = (T \otimes Y) (X \otimes Z) .

T X \otimes Y Z = (T \otimes Y) (X \otimes Z) .

F_{l} \otimes F_{r} \leavevmode \leavevmode = = (D_{l} + C_{l} (z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes (D_{r} + C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1} B_{r}) D_{l} \otimes D_{r} + D_{l} \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1} B_{r}) + (C_{l} (z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes D_{r} + (C_{l} (z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1} B_{r})

F_{l} \otimes F_{r} \leavevmode \leavevmode = = (D_{l} + C_{l} (z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes (D_{r} + C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1} B_{r}) D_{l} \otimes D_{r} + D_{l} \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1} B_{r}) + (C_{l} (z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes D_{r} + (C_{l} (z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1} B_{r})

D_{l} \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1} B_{r}) \leavevmode \leavevmode \leavevmode \leavevmode \leavevmode \leavevmode = = = = = = = D_{l} I_{m_{l}} \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1} B_{r}) (D_{l} \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1})) (I_{m_{l}} \otimes B_{r}) (D_{l} I_{m_{l}} \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1})) (I_{m_{l}} \otimes B_{r}) (D_{l} \otimes C_{r}) (I_{m_{l}} \otimes ((z_{r} I_{n_{r}} - A_{r})^{- 1})) (I_{m_{l}} \otimes B_{r}) (D_{l} \otimes I_{p_{r}}) (I_{m_{l}} \otimes C_{r}) (I_{m_{l}} \otimes ((z_{r} I_{n_{r}} - A_{r})^{- 1})) (I_{m_{l}} \otimes B_{r}) D_{l} (D_{l} \otimes I_{p_{r}}) C_{r} (I_{m_{l}} \otimes C_{r}) (z_{r} I_{m_{l} n_{r}} - A_{r} I_{m_{l}} \otimes A_{r})^{- 1} B_{r} (I_{m_{l}} \otimes B_{r}) D_{l} C_{r} (z_{r} I_{m_{l} n_{r}} - A_{r})^{- 1} B_{r}

D_{l} \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1} B_{r}) \leavevmode \leavevmode \leavevmode \leavevmode \leavevmode \leavevmode = = = = = = = D_{l} I_{m_{l}} \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1} B_{r}) (D_{l} \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1})) (I_{m_{l}} \otimes B_{r}) (D_{l} I_{m_{l}} \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1})) (I_{m_{l}} \otimes B_{r}) (D_{l} \otimes C_{r}) (I_{m_{l}} \otimes ((z_{r} I_{n_{r}} - A_{r})^{- 1})) (I_{m_{l}} \otimes B_{r}) (D_{l} \otimes I_{p_{r}}) (I_{m_{l}} \otimes C_{r}) (I_{m_{l}} \otimes ((z_{r} I_{n_{r}} - A_{r})^{- 1})) (I_{m_{l}} \otimes B_{r}) D_{l} (D_{l} \otimes I_{p_{r}}) C_{r} (I_{m_{l}} \otimes C_{r}) (z_{r} I_{m_{l} n_{r}} - A_{r} I_{m_{l}} \otimes A_{r})^{- 1} B_{r} (I_{m_{l}} \otimes B_{r}) D_{l} C_{r} (z_{r} I_{m_{l} n_{r}} - A_{r})^{- 1} B_{r}

(C_{l} (z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes D_{r} \leavevmode \leavevmode \leavevmode \leavevmode \leavevmode \leavevmode = = = = = = = (C_{l} (z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes I_{p_{r}} D_{r} (C_{l} \otimes I_{p_{r}}) (((z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes D_{r}) (C_{l} \otimes I_{p_{r}}) (((z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes I_{p_{r}} D_{r}) (C_{l} \otimes I_{p_{r}}) (((z_{l} I_{n_{l}} - A_{l})^{- 1}) \otimes I_{p_{r}}) (B_{l} \otimes D_{r}) (C_{l} \otimes I_{p_{r}}) (((z_{l} I_{n_{l}} - A_{l})^{- 1}) \otimes I_{p_{r}}) (B_{l} \otimes I_{p_{r}}) (I_{m_{l}} \otimes D_{r}) C_{l} (C_{l} \otimes I_{p_{r}}) ((z_{l} I_{n_{l} p_{r}} - A_{l} A_{l} \otimes I_{p_{r}})^{- 1} B_{l} (B_{l} \otimes I_{p_{r}}) D_{r} (I_{m_{l}} \otimes D_{r}) C_{l} ((z_{l} I_{n_{l} p_{r}} - A_{l})^{- 1} B_{l} D_{r}

(C_{l} (z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes D_{r} \leavevmode \leavevmode \leavevmode \leavevmode \leavevmode \leavevmode = = = = = = = (C_{l} (z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes I_{p_{r}} D_{r} (C_{l} \otimes I_{p_{r}}) (((z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes D_{r}) (C_{l} \otimes I_{p_{r}}) (((z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes I_{p_{r}} D_{r}) (C_{l} \otimes I_{p_{r}}) (((z_{l} I_{n_{l}} - A_{l})^{- 1}) \otimes I_{p_{r}}) (B_{l} \otimes D_{r}) (C_{l} \otimes I_{p_{r}}) (((z_{l} I_{n_{l}} - A_{l})^{- 1}) \otimes I_{p_{r}}) (B_{l} \otimes I_{p_{r}}) (I_{m_{l}} \otimes D_{r}) C_{l} (C_{l} \otimes I_{p_{r}}) ((z_{l} I_{n_{l} p_{r}} - A_{l} A_{l} \otimes I_{p_{r}})^{- 1} B_{l} (B_{l} \otimes I_{p_{r}}) D_{r} (I_{m_{l}} \otimes D_{r}) C_{l} ((z_{l} I_{n_{l} p_{r}} - A_{l})^{- 1} B_{l} D_{r}

(C_{l} (z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1} B_{r}) \leavevmode \leavevmode \leavevmode \leavevmode \leavevmode = = = = = = (C_{l} (z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l} I_{m_{l}}) \otimes (I_{p_{r}} C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1} B_{r}) (C_{l} \otimes I_{p_{r}}) ((z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1}) (I_{m_{l}} \otimes B_{r}) (C_{l} \otimes I_{p_{r}}) ((z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l} I_{m_{l}}) \otimes (I_{p_{r}} C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1}) (I_{m_{l}} \otimes B_{r}) (C_{l} \otimes I_{p_{r}}) ((z_{l} I_{n_{l}} - A_{l})^{- 1} \otimes I_{p_{r}}) (B_{l} \otimes C_{r}) (I_{m_{l}} \otimes ((z_{r} I_{n_{r}} - A_{r})^{- 1})) (I_{m_{l}} \otimes B_{r}) (C_{l} \otimes I_{p_{r}}) ((z_{l} I_{n_{l} p_{r}} - A_{l} \otimes I_{p_{r}})^{- 1} (B_{l} \otimes I_{p_{r}}) (I_{m_{l}} \otimes C_{r}) ((z_{r} I_{m_{l} n_{r}} - I_{m_{l}} \otimes A_{r})^{- 1} (I_{m_{l}} \otimes B_{r}) C_{l} ((z_{l} I_{n_{l} p_{r}} - A_{l})^{- 1} B_{l} C_{r} ((z_{r} I_{m_{l} n_{r}} - A_{r})^{- 1} B_{r}

(C_{l} (z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1} B_{r}) \leavevmode \leavevmode \leavevmode \leavevmode \leavevmode = = = = = = (C_{l} (z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l} I_{m_{l}}) \otimes (I_{p_{r}} C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1} B_{r}) (C_{l} \otimes I_{p_{r}}) ((z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l}) \otimes (C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1}) (I_{m_{l}} \otimes B_{r}) (C_{l} \otimes I_{p_{r}}) ((z_{l} I_{n_{l}} - A_{l})^{- 1} B_{l} I_{m_{l}}) \otimes (I_{p_{r}} C_{r} (z_{r} I_{n_{r}} - A_{r})^{- 1}) (I_{m_{l}} \otimes B_{r}) (C_{l} \otimes I_{p_{r}}) ((z_{l} I_{n_{l}} - A_{l})^{- 1} \otimes I_{p_{r}}) (B_{l} \otimes C_{r}) (I_{m_{l}} \otimes ((z_{r} I_{n_{r}} - A_{r})^{- 1})) (I_{m_{l}} \otimes B_{r}) (C_{l} \otimes I_{p_{r}}) ((z_{l} I_{n_{l} p_{r}} - A_{l} \otimes I_{p_{r}})^{- 1} (B_{l} \otimes I_{p_{r}}) (I_{m_{l}} \otimes C_{r}) ((z_{r} I_{m_{l} n_{r}} - I_{m_{l}} \otimes A_{r})^{- 1} (I_{m_{l}} \otimes B_{r}) C_{l} ((z_{l} I_{n_{l} p_{r}} - A_{l})^{- 1} B_{l} C_{r} ((z_{r} I_{m_{l} n_{r}} - A_{r})^{- 1} B_{r}

F_{l} \otimes F_{r} \leavevmode \leavevmode = D D_{l} \otimes D_{r} + (C_{l} D_{l} C_{r}) (((z_{l} I_{n_{l} p_{r}} - A_{l})^{- 1} 0 (z_{l} I_{n_{l} p_{r}} - A_{l})^{- 1} B_{l} C_{r} (z_{r} I_{m_{l} n_{r}} - A_{r})^{- 1} (z_{r} I_{m_{l} n_{r}} - A_{r})^{- 1}) (B_{l} D_{r} \leavevmode B_{r}) = D + (C_{l} D_{l} C_{r}) (\leavevmode (z_{l} I_{n_{l} p_{r}} 0 0 z_{r} I_{m_{l} n_{r}}) - (A_{l} 0 B_{l} C_{r} A_{r}) \leavevmode)^{- 1} (B_{l} D_{r} \leavevmode B_{r}) .

F_{l} \otimes F_{r} \leavevmode \leavevmode = D D_{l} \otimes D_{r} + (C_{l} D_{l} C_{r}) (((z_{l} I_{n_{l} p_{r}} - A_{l})^{- 1} 0 (z_{l} I_{n_{l} p_{r}} - A_{l})^{- 1} B_{l} C_{r} (z_{r} I_{m_{l} n_{r}} - A_{r})^{- 1} (z_{r} I_{m_{l} n_{r}} - A_{r})^{- 1}) (B_{l} D_{r} \leavevmode B_{r}) = D + (C_{l} D_{l} C_{r}) (\leavevmode (z_{l} I_{n_{l} p_{r}} 0 0 z_{r} I_{m_{l} n_{r}}) - (A_{l} 0 B_{l} C_{r} A_{r}) \leavevmode)^{- 1} (B_{l} D_{r} \leavevmode B_{r}) .

D_{l} \otimes D_{r} = (D_{l} I_{m_{l}}) \otimes (I_{p_{r}} D_{r}) = D_{l} (D_{l} \otimes I_{p_{r}}) D_{r} (I_{m_{l}} \otimes D_{r}) = D_{l} D_{r} = D .

D_{l} \otimes D_{r} = (D_{l} I_{m_{l}}) \otimes (I_{p_{r}} D_{r}) = D_{l} (D_{l} \otimes I_{p_{r}}) D_{r} (I_{m_{l}} \otimes D_{r}) = D_{l} D_{r} = D .

R_{F_{l}\otimes F_{r}}=\footnotesize\left(\begin{array}[]{cc|c}\mathbf{A_{l}}&\mathbf{B_{l}}\mathbf{C_{r}}&\mathbf{B_{l}}\mathbf{D_{r}}\\ 0&\mathbf{A_{r}}&\mathbf{B_{r}}\\ \hline\cr\mathbf{C_{l}}&\mathbf{D_{l}}\mathbf{C_{r}}&\mathbf{D_{l}}\mathbf{D_{r}}\end{array}\right)=\footnotesize\left(\begin{array}[]{c|c}\mathbf{A_{o}}&\mathbf{B_{o}}\\ \hline\cr\mathbf{C_{o}}&\mathbf{D}\end{array}\right)={\mathbf{R}},

R_{F_{l}\otimes F_{r}}=\footnotesize\left(\begin{array}[]{cc|c}\mathbf{A_{l}}&\mathbf{B_{l}}\mathbf{C_{r}}&\mathbf{B_{l}}\mathbf{D_{r}}\\ 0&\mathbf{A_{r}}&\mathbf{B_{r}}\\ \hline\cr\mathbf{C_{l}}&\mathbf{D_{l}}\mathbf{C_{r}}&\mathbf{D_{l}}\mathbf{D_{r}}\end{array}\right)=\footnotesize\left(\begin{array}[]{c|c}\mathbf{A_{o}}&\mathbf{B_{o}}\\ \hline\cr\mathbf{C_{o}}&\mathbf{D}\end{array}\right)={\mathbf{R}},

n_{l} p_{r} + m_{l} n_{r} \leavevmode .

n_{l} p_{r} + m_{l} n_{r} \leavevmode .

{\mathbf{R}}=\footnotesize\left(\begin{array}[]{cc|c}\mathbf{A_{l}}&\mathbf{B_{l}}\mathbf{C_{r}}&\mathbf{B_{l}}\mathbf{D_{r}}\\ 0&\mathbf{A_{r}}&\mathbf{B_{r}}\\ \hline\cr\mathbf{C_{l}}&\mathbf{D_{l}}\mathbf{C_{r}}&\mathbf{D}\end{array}\right)=\left(\begin{smallmatrix}\mathbf{A_{l}}&0&\mathbf{B_{l}}\\ 0&I_{m_{l}n_{r}}&0\\ \mathbf{C_{l}}&0&\mathbf{D_{l}}\end{smallmatrix}\right)\left(\begin{smallmatrix}I_{n_{l}p_{r}}&0&0\\ 0&\mathbf{A_{r}}&\mathbf{B_{r}}\\ 0&\mathbf{C_{r}}&\mathbf{D_{r}}\end{smallmatrix}\right).

{\mathbf{R}}=\footnotesize\left(\begin{array}[]{cc|c}\mathbf{A_{l}}&\mathbf{B_{l}}\mathbf{C_{r}}&\mathbf{B_{l}}\mathbf{D_{r}}\\ 0&\mathbf{A_{r}}&\mathbf{B_{r}}\\ \hline\cr\mathbf{C_{l}}&\mathbf{D_{l}}\mathbf{C_{r}}&\mathbf{D}\end{array}\right)=\left(\begin{smallmatrix}\mathbf{A_{l}}&0&\mathbf{B_{l}}\\ 0&I_{m_{l}n_{r}}&0\\ \mathbf{C_{l}}&0&\mathbf{D_{l}}\end{smallmatrix}\right)\left(\begin{smallmatrix}I_{n_{l}p_{r}}&0&0\\ 0&\mathbf{A_{r}}&\mathbf{B_{r}}\\ 0&\mathbf{C_{r}}&\mathbf{D_{r}}\end{smallmatrix}\right).

F_{a} (z_{a}) \otimes F_{b} (z_{b}) \otimes F_{c} (z_{c}) \dots

F_{a} (z_{a}) \otimes F_{b} (z_{b}) \otimes F_{c} (z_{c}) \dots

R_{F}=\footnotesize\left(\begin{array}[]{c|c}A&B\\ \hline\cr C&D\end{array}\right),

R_{F}=\footnotesize\left(\begin{array}[]{c|c}A&B\\ \hline\cr C&D\end{array}\right),

R_{F^{-1}}=\footnotesize\left(\begin{array}[]{c|c}A^{\times}&B^{\times}\\ \hline\cr C^{\times}&D^{\times}\end{array}\right)=\footnotesize\left(\begin{array}[]{c|c}A-BD^{-1}C&-BD^{-1}\\ \hline\cr D^{-1}C&D^{-1}\end{array}\right).

R_{F^{-1}}=\footnotesize\left(\begin{array}[]{c|c}A^{\times}&B^{\times}\\ \hline\cr C^{\times}&D^{\times}\end{array}\right)=\footnotesize\left(\begin{array}[]{c|c}A-BD^{-1}C&-BD^{-1}\\ \hline\cr D^{-1}C&D^{-1}\end{array}\right).

m_{l} = p_{r} and p_{l} = m_{r}

m_{l} = p_{r} and p_{l} = m_{r}

\begin{matrix}R_{(F_{l}F_{r})^{-1}}=\footnotesize\left(\begin{array}[]{cc|c}A_{l}^{\times}&0&B_{l}^{\times}\\ B_{r}^{\times}C_{l}^{\times}&A_{r}^{\times}&B_{r}^{\times}D_{l}^{-1}\\ \hline\cr D_{r}^{-1}C_{l}^{\times}&C_{r}^{\times}&D_{r}^{-1}D_{l}^{-1}\end{array}\right)=\left(\begin{smallmatrix}I_{n_{l}}&0&0\\ 0&A_{r}^{\times}&B_{r}^{\times}\\ 0&C_{r}^{\times}&D_{r}^{-1}\end{smallmatrix}\right)\left(\begin{smallmatrix}A_{l}^{\times}&0&B_{l}^{\times}\\ 0&I_{n_{r}}&0\\ C_{l}^{\times}&0&D_{l}^{-1}\end{smallmatrix}\right).\end{matrix}

\begin{matrix}R_{(F_{l}F_{r})^{-1}}=\footnotesize\left(\begin{array}[]{cc|c}A_{l}^{\times}&0&B_{l}^{\times}\\ B_{r}^{\times}C_{l}^{\times}&A_{r}^{\times}&B_{r}^{\times}D_{l}^{-1}\\ \hline\cr D_{r}^{-1}C_{l}^{\times}&C_{r}^{\times}&D_{r}^{-1}D_{l}^{-1}\end{array}\right)=\left(\begin{smallmatrix}I_{n_{l}}&0&0\\ 0&A_{r}^{\times}&B_{r}^{\times}\\ 0&C_{r}^{\times}&D_{r}^{-1}\end{smallmatrix}\right)\left(\begin{smallmatrix}A_{l}^{\times}&0&B_{l}^{\times}\\ 0&I_{n_{r}}&0\\ C_{l}^{\times}&0&D_{l}^{-1}\end{smallmatrix}\right).\end{matrix}

m_{l} m_{r} = p_{l} p_{r},

m_{l} m_{r} = p_{l} p_{r},

m_{l} = p_{l} m_{r} = p_{r}

m_{l} = p_{l} m_{r} = p_{r}

R_{(F_{l} \otimes F_{r})^{- 1}} = ((A_{l} \otimes I_{p_{r}}) - (B_{l} \otimes D_{r}) (D_{l} \otimes D_{r})^{- 1} (C_{l} \otimes I_{p_{r}}) - (I_{m_{l}} \otimes B_{r}) (D_{l} \otimes D_{r})^{- 1} (C_{l} \otimes I_{p_{r}}) (D_{l} \otimes D_{r})^{- 1} (C_{l} \otimes I_{p_{r}}) B_{l} \otimes C_{r} - (B_{l} \otimes D_{r}) (D_{l} \otimes D_{r})^{- 1} (D_{l} \otimes C_{r}) (I_{m_{l}} \otimes A_{r}) - (I_{m_{l}} \otimes B_{r}) (D_{l} \otimes D_{r})^{- 1} (D_{l} \otimes C_{r}) (D_{l} \otimes D_{r})^{- 1} (D_{l} \otimes C_{r}) - (B_{l} \otimes D_{r}) (D_{l} \otimes D_{r})^{- 1} - (I_{m_{l}} \otimes B_{r}) (D_{l} \otimes D_{r})^{- 1} (D_{l} \otimes D_{r})^{- 1}) .

R_{(F_{l} \otimes F_{r})^{- 1}} = ((A_{l} \otimes I_{p_{r}}) - (B_{l} \otimes D_{r}) (D_{l} \otimes D_{r})^{- 1} (C_{l} \otimes I_{p_{r}}) - (I_{m_{l}} \otimes B_{r}) (D_{l} \otimes D_{r})^{- 1} (C_{l} \otimes I_{p_{r}}) (D_{l} \otimes D_{r})^{- 1} (C_{l} \otimes I_{p_{r}}) B_{l} \otimes C_{r} - (B_{l} \otimes D_{r}) (D_{l} \otimes D_{r})^{- 1} (D_{l} \otimes C_{r}) (I_{m_{l}} \otimes A_{r}) - (I_{m_{l}} \otimes B_{r}) (D_{l} \otimes D_{r})^{- 1} (D_{l} \otimes C_{r}) (D_{l} \otimes D_{r})^{- 1} (D_{l} \otimes C_{r}) - (B_{l} \otimes D_{r}) (D_{l} \otimes D_{r})^{- 1} - (I_{m_{l}} \otimes B_{r}) (D_{l} \otimes D_{r})^{- 1} (D_{l} \otimes D_{r})^{- 1}) .

\begin{matrix}R_{(F_{l}\otimes F_{r})^{-1}}&=&\footnotesize\left(\begin{array}[]{cc|c}\left(A_{l}-B_{l}D_{l}^{-1}C_{l}\right)\otimes I_{p_{r}}&0&\left(-B_{l}D_{l}^{-1}\right)\otimes I_{p_{r}}\\ \left(D_{l}^{-1}C_{l}\right)\otimes\left(-B_{r}D_{r}^{-1}\right)&I_{m_{l}}\otimes\left(A_{r}-B_{r}D_{r}^{-1}C_{r}\right)&D_{l}^{-1}\otimes\left(-B_{r}D_{r}^{-1}\right)\\ \hline\cr D_{l}^{-1}C_{l}\otimes D_{r}^{-1}&I_{m_{l}}\otimes D_{r}^{-1}C_{r}&\left(D_{l}\otimes D_{r}\right)^{-1}\end{array}\right)\\ \leavevmode\nobreak\ \\ \leavevmode\nobreak\ &=&\footnotesize\left(\begin{array}[]{cc|c}A_{l}^{\times}{\otimes}I_{p_{r}}&0&B_{l}^{\times}\otimes I_{p_{r}}\\ C_{l}^{\times}\otimes B_{r}^{\times}&I_{m_{l}}\otimes A_{r}^{\times}&D_{l}^{-1}\otimes B_{r}^{\times}\\ \hline\cr C_{l}^{\times}\otimes D_{r}^{-1}&I_{m_{l}}\otimes C_{r}^{\times}&\left(D_{l}\otimes D_{r}\right)^{-1}\end{array}\right)\\ \leavevmode\nobreak\ \\ \leavevmode\nobreak\ &=&\footnotesize\left(\begin{array}[]{cc|c}A_{l}^{\times}\otimes I_{p_{r}}&0&\left(B_{l}^{\times}\otimes I_{p_{r}}\right)\\ \left(I_{n_{l}}\otimes B_{r}^{\times}\right)\left(C_{l}^{\times}\otimes I_{p_{r}}\right)&I_{m_{l}}\otimes A_{r}^{\times}&\left(I_{m_{l}}\otimes B_{r}^{\times}\right)\left(D_{l}^{-1}\otimes I_{n_{r}}\right)\\ \hline\cr\left(I_{n_{l}}\otimes D_{r}^{-1}\right)\left(C_{l}^{\times}\otimes I_{p_{r}}\right)&I_{m_{l}p_{r}}\otimes\left(I_{m_{l}}\otimes C_{r}^{\times}\right)&\left(I_{m_{l}}\otimes D_{r}^{-1}\right)\left(D_{l}^{-1}\otimes I_{p_{r}}\right)\end{array}\right)\\ \leavevmode\nobreak\ \\ \leavevmode\nobreak\ &=&\footnotesize\left(\begin{array}[]{cc|c}\mathbf{A_{l}^{\times}}&0&\mathbf{B_{l}^{\times}}\\ \mathbf{B_{r}^{\times}}\mathbf{C_{l}^{\times}}&\mathbf{A_{r}^{\times}}&\mathbf{B_{r}^{\times}}\mathbf{D_{l}}^{-1}\\ \hline\cr\mathbf{D_{r}^{-1}}\mathbf{C_{l}^{\times}}&\mathbf{C_{r}^{\times}}&\mathbf{D_{r}^{-1}}\mathbf{D_{l}^{-1}}\end{array}\right)=\footnotesize\left(\begin{array}[]{c|c}\mathbf{{A_{o}}^{\times}}&\mathbf{{B_{o}}^{\times}}\\ \hline\cr\mathbf{{C_{o}}^{\times}}&\mathbf{D^{\times}}\end{array}\right)=\mathbf{R^{\times}},\end{matrix}

\begin{matrix}R_{(F_{l}\otimes F_{r})^{-1}}&=&\footnotesize\left(\begin{array}[]{cc|c}\left(A_{l}-B_{l}D_{l}^{-1}C_{l}\right)\otimes I_{p_{r}}&0&\left(-B_{l}D_{l}^{-1}\right)\otimes I_{p_{r}}\\ \left(D_{l}^{-1}C_{l}\right)\otimes\left(-B_{r}D_{r}^{-1}\right)&I_{m_{l}}\otimes\left(A_{r}-B_{r}D_{r}^{-1}C_{r}\right)&D_{l}^{-1}\otimes\left(-B_{r}D_{r}^{-1}\right)\\ \hline\cr D_{l}^{-1}C_{l}\otimes D_{r}^{-1}&I_{m_{l}}\otimes D_{r}^{-1}C_{r}&\left(D_{l}\otimes D_{r}\right)^{-1}\end{array}\right)\\ \leavevmode\nobreak\ \\ \leavevmode\nobreak\ &=&\footnotesize\left(\begin{array}[]{cc|c}A_{l}^{\times}{\otimes}I_{p_{r}}&0&B_{l}^{\times}\otimes I_{p_{r}}\\ C_{l}^{\times}\otimes B_{r}^{\times}&I_{m_{l}}\otimes A_{r}^{\times}&D_{l}^{-1}\otimes B_{r}^{\times}\\ \hline\cr C_{l}^{\times}\otimes D_{r}^{-1}&I_{m_{l}}\otimes C_{r}^{\times}&\left(D_{l}\otimes D_{r}\right)^{-1}\end{array}\right)\\ \leavevmode\nobreak\ \\ \leavevmode\nobreak\ &=&\footnotesize\left(\begin{array}[]{cc|c}A_{l}^{\times}\otimes I_{p_{r}}&0&\left(B_{l}^{\times}\otimes I_{p_{r}}\right)\\ \left(I_{n_{l}}\otimes B_{r}^{\times}\right)\left(C_{l}^{\times}\otimes I_{p_{r}}\right)&I_{m_{l}}\otimes A_{r}^{\times}&\left(I_{m_{l}}\otimes B_{r}^{\times}\right)\left(D_{l}^{-1}\otimes I_{n_{r}}\right)\\ \hline\cr\left(I_{n_{l}}\otimes D_{r}^{-1}\right)\left(C_{l}^{\times}\otimes I_{p_{r}}\right)&I_{m_{l}p_{r}}\otimes\left(I_{m_{l}}\otimes C_{r}^{\times}\right)&\left(I_{m_{l}}\otimes D_{r}^{-1}\right)\left(D_{l}^{-1}\otimes I_{p_{r}}\right)\end{array}\right)\\ \leavevmode\nobreak\ \\ \leavevmode\nobreak\ &=&\footnotesize\left(\begin{array}[]{cc|c}\mathbf{A_{l}^{\times}}&0&\mathbf{B_{l}^{\times}}\\ \mathbf{B_{r}^{\times}}\mathbf{C_{l}^{\times}}&\mathbf{A_{r}^{\times}}&\mathbf{B_{r}^{\times}}\mathbf{D_{l}}^{-1}\\ \hline\cr\mathbf{D_{r}^{-1}}\mathbf{C_{l}^{\times}}&\mathbf{C_{r}^{\times}}&\mathbf{D_{r}^{-1}}\mathbf{D_{l}^{-1}}\end{array}\right)=\footnotesize\left(\begin{array}[]{c|c}\mathbf{{A_{o}}^{\times}}&\mathbf{{B_{o}}^{\times}}\\ \hline\cr\mathbf{{C_{o}}^{\times}}&\mathbf{D^{\times}}\end{array}\right)=\mathbf{R^{\times}},\end{matrix}

A_{l}^{\times} := A_{l}^{\times} \otimes I_{p_{r}} B_{l}^{\times} := B_{l}^{\times} \otimes I_{p_{r}} C_{l}^{\times} := C_{l}^{\times} \otimes I_{p_{r}} D_{l}^{- 1} = D_{l}^{- 1} \otimes I_{p_{r}} A_{r}^{\times} := I_{m_{l}} \otimes A_{r}^{\times} B_{r}^{\times} := I_{m_{l}} \otimes B_{r}^{\times} C_{r}^{\times} := I_{m_{l}} \otimes C_{r}^{\times} D_{r}^{- 1} = I_{m_{l}} \otimes D_{r}^{- 1} .

A_{l}^{\times} := A_{l}^{\times} \otimes I_{p_{r}} B_{l}^{\times} := B_{l}^{\times} \otimes I_{p_{r}} C_{l}^{\times} := C_{l}^{\times} \otimes I_{p_{r}} D_{l}^{- 1} = D_{l}^{- 1} \otimes I_{p_{r}} A_{r}^{\times} := I_{m_{l}} \otimes A_{r}^{\times} B_{r}^{\times} := I_{m_{l}} \otimes B_{r}^{\times} C_{r}^{\times} := I_{m_{l}} \otimes C_{r}^{\times} D_{r}^{- 1} = I_{m_{l}} \otimes D_{r}^{- 1} .

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TopicsMatrix Theory and Algorithms · Tensor decomposition and applications · Numerical Methods and Algorithms

Full text

realization of tensor-product and of

Tensor-Factorization of rational functions

Daniel Alpay

(DA) Faculty of Mathematics, Physics, and Computation

Schmidt College of Science and Technology

Chapman University

One University Drive Orange, California 92866

USA

[email protected]

and

Izchak Lewkowicz

(IL) Department of electrical engineering Ben-Gurion University of the Negev

P.O.B. 653

Beer-Sheva, 84105

Israel

[email protected]

Abstract.

We here first study the state space realization of a tensor-product of a pair of rational functions. At the expense of “inflating” the dimensions, we recover the classical expressions for realization of a regular product of rational functions. Then, under an additional assumption that the limit at infinity of a given rational function exists and is equal to identity, an explicit formula for a tensor-factorization of this function, is introduced.

Daniel Alpay thanks the Foster G. and Mary McGaw Professorship in Mathematical Sciences, which supported this research.

today

1. Introduction

The problem of minimal factorization of matrix-valued rational functions of one complex variable has along history; see for instance [1, 2, 6]. Less known seems to be the counterpart of this problem when matrix product is replaced by tensor product. More precisely, we study the following two problems: First, given two rational matrix-valued functions $R_{1}$ and $R_{2}$ analytic at infinity, write a realization of the tensor product $R_{1}\otimes R_{2}$ in terms of realizations of $R_{1}$ and $R_{2}$ . Next, given a matrix-valued rational function $R$ analytic at infinity, find its representations as $R_{1}\otimes R_{2}$ where $R_{1}$ and $R_{2}$ are rational and analytic at infinity.

To provide some motivation we note the following. Tensor products play an important role in mathematics and quantum mechanics. In the latter case, a first example (see e.g. [4, p. 162]) is the product of two wave functions, each belonging to a given Hilbert space, which belongs to the tensor product of the given Hilbert spaces; see e.g. [8, Proposition 6.2, p. 111] for the latter. Another example is the case of quantum states (positive matrices with trace equal to $1$ ; see e.g. [9]). Given two states $M_{1}\in\mathbb{C}^{N_{1}\times N_{1}}$ and $M_{2}\in\mathbb{C}^{N_{2}\times N_{2}}$ , of possibly different sizes, the tensor product $M_{1}\otimes M_{2}$ is still a state. Note that if $M=M_{1}\otimes M_{2}$ , one can recover $M_{1}$ and $M_{2}$ uniquely via the formula

[TABLE]

where $f_{1},\ldots,f_{N_{2}}$ denotes an orthonormal basis for $\mathbb{C}^{N_{2}}$ , and similarly for $M_{2}$ ,

[TABLE]

where now $e_{1},\ldots,e_{N_{1}}$ is an orthonormal basis for $\mathbb{C}^{N_{1}}$ . See e.g. [9, eq. (9.2.1) p. 97].

If one starts from an arbitrary state $M\in\mathbb{C}^{N_{1}N_{2}\times N_{1}N_{2}}$ the matrices defined by (1.1) and (1.2) will be states, called marginal states, but their tensor product need not be equal to $M$ .

One can consider similar problems in the setting of functions. We focus the discussion on rational functions. If $R(z)$ is a $\mathbb{C}^{N\times N}$ -valued rational function and if $N=N_{1}N_{2}$ , formulas (1.1) and (1.2) now define two rational functions $R_{A}$ and $R_{B}$ , respectively $\mathbb{C}^{N_{1}\times N_{1}}$ and $\mathbb{C}^{N_{1}\times N_{1}}$ -valued, via

[TABLE]

If $R=R_{1}\otimes R_{2}$ where $R_{1}$ is $\mathbb{C}^{N_{1}\times N_{1}}$ -valued and $R_{2}$ is $\mathbb{C}^{N_{2}\times N_{2}}$ -valued, then these equations can be rewritten as

[TABLE]

and so these equations basically solve the tensor factorization problem.

The purpose of this work is in a somewhat different direction; we would like to express both tensor multiplication and tensor factorization of matrix-valued rational functions using state space representations.

In the rest of this section we cite some known results. Let $z_{l}$ , $z_{r}$ (the subscript stands for “left” and “right”) be a pair of complex variables, and let $F_{l}(z_{l})$ , $F_{r}(z_{r})$ be a pair of $p_{l}\times m_{l}$ , $p_{r}\times m_{r}$ -valued rational functions, respectively. Assume that neither has poles at infinity and denote by $n_{l}$ , $n_{r}$ the respective McMillan degrees. Thus, one can write the rational functions and the respective realization as

[TABLE]

Recall that whenever $m_{l}=p_{r}$ the product $F_{l}(z_{l})F_{r}(z_{r})$ is well-defined and its realization is given by111 Strictly speaking, in the references it was formulated for $z_{l}=z_{r}=z$ i.e. for $F_{l}(z)F_{r}(z)$ (see e.g. [3, Section 2.5])

[TABLE]

in the sense that

[TABLE]

If $z_{l}=z_{r}$ the sought realization in (1.6) is of McMillan degree

[TABLE]

when minimal (roughly speaking when there is no pole-zero cancelation). We next address ourselves to the tensor product222In matrix theory circles known as the “Kronecker product ”, see e.g. [7, Section 4.2]. of $F_{l}(z_{l})$ and $F_{r}(z_{r})$ , resulting in $F_{l}\otimes F_{r}$ , a $p_{l}p_{r}\times m_{l}m_{r}$ -valued rational function. Tensor product of rational functions is discussed in [5, Section 5.2].

So far for known results. In the next section we focus on $R_{F_{l}\otimes F_{r}}$ , the state space realization of $F_{l}\otimes F_{r}$ . In Section 3 we set the framework for the main result, which is the factorization result presented in Section 4.

2. Realization of a tensor-product of rational functions

We start with technicalities: We denote by boldface characters, “inflated version” of the original ones, i.e.

[TABLE]

We then show that at the expense of “inflating” the dimensions one can replace a tensor product by a usual product.

Proposition 2.1.

Let $F_{l}(z_{r})$ , $F_{r}(z_{r})$ be a pair of $p_{l}\times m_{l}$ , $p_{r}\times m_{r}$ -valued rational functions, of McMillan degree $n_{l}$ , $n_{r}$ , respectively, whose realization is given in Eq. (1.5). Following Eqs. (1.6) and (2.8), one has that,

[TABLE]

In order to go into details we shall repeatedly use the fact, see e.g. [7, Lemma 4.2.10], that for matrices $T\in{\mathbb{C}}^{n\times m}$ , $X\in{\mathbb{C}}^{m\times l}$ , $Y\in{\mathbb{C}}^{l\times p}$ , $Z\in{\mathbb{C}}^{p\times q}$ one has that

[TABLE]

We now explicitly compute the tensor product of $F_{l}(z_{l})$ and $F_{r}(z_{r})$ ,

[TABLE]

We next separately examine each block

[TABLE]

Thus, one can write

[TABLE]

Note that in particular

[TABLE]

The realization of $F_{l}(z_{l})\otimes F_{r}(z_{r})$ can be compactly written as

[TABLE]

which is indeed in form of (1.6), (2.8). If $z_{l}=z_{r}$ and there is no pole-zero cancelation, the sought realization in (2.11) is of McMillan degree

[TABLE]

Note now that in a way similar to (1.6), one can factorize the realization in (2.11) as follows,

[TABLE]

We conclude this section by pointing out that Proposition 2.1 can be easily extended to more elaborate cases like

[TABLE]

3. Realization of the inverse of a tensor product of rational functions

For future reference, in this section we examine the realization of the inverse of rational functions of the form $F_{l}(z_{l})\otimes F_{r}(z_{r})$ studied in the previous section.

We first recall, see e.g. [3, Theorem 2.4], in the realization of the inverse a rational function: Namely if

[TABLE]

is a realization of a square matrix-valued rational function $F(z)$ , then whenever $D$ is non-singular, $\left(F(z)\right)^{-1}$ is well-defined almost everywhere, and a corresponding realization is given by,

[TABLE]

Next, whenever the above $F_{l}(z)$ and $F_{r}(z)$ are so that

[TABLE]

the product $F_{l}(z)F_{r}(z)$ is square, and whenever $D_{l}D_{r}$ is non-singular333this implies that $m_{l}=p_{r}\geq{\rm rank}(D_{l}D_{r})=p_{l}=m_{r}$ ., $\left(F_{l}(z)F_{r}(z)\right)^{-1}$ is well-defined almost everywhere, and by combining (1.6) together with (3.13) a corresponding realization is given by

[TABLE]

Similarly, whenever

[TABLE]

the rational function $F_{l}(z)\otimes F_{r}(z)$ is square and if $D_{l}\otimes D_{r}=\mathbf{D_{l}}\mathbf{D_{r}}=\mathbf{D}$ is non-singular, then $D_{l}$ , $D_{r}$ are square, i.e.

[TABLE]

and non-singular, see e.g. [7, Theorem 4.2.15]. Thus, we shall denote hereafter by $m_{l}\times m_{l}$ , $m_{r}\times m_{r}$ the dimensions of $F_{l}$ , $F_{r}$ , respectively.

Under these conditions, the $m_{l}m_{r}\times m_{l}m_{r}$ -valued rational function, $\left(F_{l}(z)\otimes F_{r}(z)\right)^{-1}$ is almost everywhere defined. (2.11), we next compute the realization of $(F_{l}\otimes F_{r})^{-1}$ ,

[TABLE]

Taking into account the fact that $D_{l}$ and $D_{r}$ are square and non-singular, the realization $R_{(F_{l}\otimes F_{r})^{-1}}$ takes the form

[TABLE]

where the boldface entries are given by

[TABLE]

One can conclude that

[TABLE]

and in a way similar to (2.12), one can factorize the above realization as follows,

[TABLE]

4. Tensor-factorization of rational functions

We now address a more challenging question: Given $\mathbf{F}(z)$ and $\left(\mathbf{F}(z)\right)^{-1}$ (assuming that $\det\mathbf{F}(z)\not\equiv 0$ ), under what conditions and how, can it be “tensor-factorized” to some $F_{l}(z)$ and $F_{r}(z)$ , namely the following relation holds,

[TABLE]

Note that here, we confine the discussion to a single complex variable, i.e. $z_{l}=z_{r}=z$ .

Note also that if (4.1) holds, this is true up to complex scaling i.e.,

[TABLE]

We shall use this degree of freedom in the sequel.

We next recall in the following fact from matrix theory.

Let $\Pi_{\alpha}$ , $\Pi_{\beta}$ be a pair of supporting projections of the space $\mathbb{C}^{(\alpha+\beta)\times(\alpha+\beta)}$ , i.e.

[TABLE]

Such a pair of projections can be obtained by partitioning an arbitrary non-singular $T\in\mathbb{C}^{(\alpha+\beta)\times(\alpha+\beta)}$ as follows.

[TABLE]

By using an isometry-like relation, we next offer a simple way to “deflate” matrix dimensions.

Observation 4.1.

Given $M\in\mathbb{C}^{s\times q}$ , denote

[TABLE]

For arbitrary $u\in\mathbb{C}^{p}$ , $v\in\mathbb{C}^{m}$ normalized so that $u^{*}u=1$ , $v^{*}v=1$ , one has that

[TABLE]

Indeed, by twice applying (2.10) one obtains,

[TABLE]

We next apply the last observation to the variables here.

Corollary 4.2.

For $u\in\mathbb{C}^{p_{r}}$ , $v\in\mathbb{C}^{m_{l}}$ , normalized so that $u^{*}u=1$ and $v^{*}v=1$ , the boldface characters in (2.8) satisfy

[TABLE]

We now return to the problem of “tensor-factorization” in (4.1). We note that in place of $\mathbf{R}$ in (2.11) and $\mathbf{R^{\times}}$ in (3.16), the realization arrays associated with $\mathbf{F}$ and $\mathbf{F^{-1}}$ , are known only up to a coordinate transformation, i.e. there exists, a non-singular matrix $T\in{\mathbb{C}}^{(n_{l}p_{r}+m_{l}n_{r})\times(n_{l}p_{r}+m_{l}n_{r})}$ namely in (4.2) and (4.3)

[TABLE]

so that the actual realization array is given by

[TABLE]

and

[TABLE]

As in reality, the specific coordinate transformation, $T$ in (4.4) and (4.5) is unknown one can conclude that to extract $F_{l}(z)$ and $F_{r}(z)$ from (4.1) along with the realization arrays in (4.4), (4.5), additional conditions are needed.

Theorem 4.3.

Let ${\mathbf{F}}(z)$ be a given square matrix-valued rational function. Assume that

[TABLE]

Let $\leavevmode\nobreak\ \footnotesize\left(\begin{array}[]{c|c}\mathbf{A}&\mathbf{B}\\ \hline\cr\mathbf{C}&I\end{array}\right),\leavevmode\nobreak\$ see (4.4), and $\leavevmode\nobreak\ \footnotesize\left(\begin{array}[]{c|c}\mathbf{A^{\times}}&\mathbf{B^{\times}}\\ \hline\cr\mathbf{C^{\times}}&I\end{array}\right).$ see (4.5), be realizations of $\mathbf{F}(z)$ and of $\left(\mathbf{F}(z)\right)^{-1}$ , respectively.

Substituting in (4.2), $\alpha=n_{l}m_{r}$ and $\beta=m_{l}n_{r}$ , assume also that there exists a pair of supporting projection to $\mathbb{C}^{n_{l}m_{r}+m_{l}n_{r}}$ denoted by $\Pi_{n_{l}m_{r}}$ and $\Pi_{m_{l}n_{r}}$ so that

[TABLE]

Following the definition of the projections ${\Pi}_{n_{l}m_{r}}$ and $\leavevmode\nobreak\ {\Pi}_{m_{l}n_{r}}$ , see (4.3) and (4.6), along with Corollary 4.2, for arbitrary $u\in\mathbb{C}^{m_{r}}$ , $v\in\mathbb{C}^{m_{l}}$ , normalized so that $u^{*}u=1$ and $v^{*}v=1$ , we find it convenient to introduce the following related projections444note that $\hat{\Pi}_{n_{l}m_{r}}{\Pi}_{n_{l}m_{r}}=\hat{\Pi}_{n_{l}m_{r}}{\Pi}_{n_{l}m_{r}}=\hat{\Pi}_{n_{l}m_{r}}$ and $\leavevmode\nobreak\ \hat{\Pi}_{m_{l}n_{r}}{\Pi}_{m_{l}n_{r}}={\Pi}_{m_{l}n_{r}}\hat{\Pi}_{m_{l}n_{r}}=\hat{\Pi}_{m_{l}n_{r}}$ .,

[TABLE]

Then, using (2.11) and (4.4), one can take in (4.1) $\mathbf{F}=F_{l}\otimes F_{r}$ where,

[TABLE]

Proof : First, recall (see Section 3) that the assumption that $D_{l}\otimes D_{r}=\mathbf{D_{l}}\mathbf{D_{r}}=\mathbf{D}$ is square non-singular, it implies that both $D_{l}$ and $D_{r}$ are square non-singular. We shall thus denote the dimensions of $F_{l}$ and $F_{r}$ , by $m_{l}\times m_{l}$ and $m_{r}\times m_{r}$ , respectively.

The assumption here that $\mathbf{D}=I_{m_{l}m_{r}}\leavevmode\nobreak\$ implies (see e.e. [7, Theorem 4.2.12]) that

[TABLE]

As already mentioned after (4.1), to simplify the exposition we shall take $c=1$ .

Next, let $T$ in (4.3), (4.4), (4.5) be the same so that the supporting projections are $\Pi_{n_{l}m_{r}}$ and $\Pi_{m_{l}n_{r}}$ . Next note that substituting (2.11), (3.16), (4.4) and (4.5) in condition (4.6) yields,

[TABLE]

and thus in the sequel we shall use the two upper relations, i.e.

[TABLE]

We are now ready to recover $F_{l}(z)$ ,

[TABLE]

Similarly, for $F_{r}(z)$

[TABLE]

∎

Remark 4.4.

At first sight, the assumptions in Theorem 4.3 seem very restrictive. For persective recall that to factorize a given rational function $F(z)$ to $F(z)=F_{l}(z)F_{r}(z)$ , the assumptions are virtually the same555There they only assume $D$ is square non-singular, but then only $F_{l}(z)D_{r}$ and $D_{l}F_{r}(z)$ are obtained., see [3, Section 2.5]).

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realization of tensor-product and of

Abstract.

1. Introduction

2. Realization of a tensor-product of rational functions

Proposition 2.1**.**

3. Realization of the inverse of a tensor product of rational functions

4. Tensor-factorization of rational functions

Observation 4.1**.**

Corollary 4.2**.**

Theorem 4.3**.**

Remark 4.4**.**

Proposition 2.1.

Observation 4.1.

Corollary 4.2.

Theorem 4.3.

Remark 4.4.