Fiedler Linearizations of Rectangular Rational Matrix Functions

Abstract

Linearization is a standard approach in the computation of eigenvalues, eigenvectors and invariant subspaces of matrix polynomials and rational matrix valued functions. An important source of linearizations are the so called Fiedler linearizations, which are generalizations of the classical companion forms. In this paper the concept of Fiedler linearization is extended from square regular to rectangular rational matrix valued functions. The approach is applied to Rosenbrock functions arising in mathematical system theory.

Appendix

So far we have only considered the case that $d_A\ge d_D$ in the polynomial degrees. For completeness in this appendix, we present the analogous results and algorithms for the case $d_{A} < d_{D}$.

Based on the construction in Algorithm 4 we have the following.

Lemma 6.1

Let $S(\lambda )$ be as in (1.5) with $A(\lambda ) = \sum \nolimits _{i=0}^{d_A}\lambda ^{i}A_i\in \mathbb C[\lambda ]^{n\times n}$, $D(\lambda ) = \sum \nolimits _{i=0}^{d_D}\lambda ^{i}D_{i}\in \mathbb C[\lambda ]^{p\times m}$ and suppose that $d_A< d_D$. For a bijection $\sigma$ Algorithm 4 constructs a sequence of matrices $\{{\mathbb {W}}_0, {\mathbb {W}}_1, \ldots , {\mathbb {W}}_{d_{D}-2}\},$ where each matrix ${\mathbb {W}}_i$ for $i = 1, 2, \ldots , d_{D}-2$ is partitioned into blocks in such a way that the blocks of ${\mathbb {W}}_{i-1}$ are blocks of ${\mathbb {W}}_i.$

Then

• (a) The size of ${\mathbb {W}}_i$ for $i = 0:d_{A}-2$ is $\left[ \left( n+n c(\sigma (0:i)) + n \mathfrak {i}(\sigma (0:i))\right) + \left( p+p c(\sigma (0:i)) + m \mathfrak {i}(\sigma (0:i))\right) \right] \times \left[ \left( n + n c(\sigma (0:i)) + n \mathfrak {i}(\sigma (0:i))\right) + \left( m + p c(\sigma (0:i)) + m \mathfrak {i}(\sigma (0:i))\right) \right]$ and for $i = d_{A}-1: d_{D}-2$ is $\left[ d_{A}n + \left( p+p c(\sigma (0:i)) + m \mathfrak {i}(\sigma (0:i))\right) \right] \times \left[ d_{A}n + \left( m + p c(\sigma (0:i)) \right. \right. \left. \left. + m \mathfrak {i}(\sigma (0:i))\right) \right] .$

• (b) The (1, 1) diagonal block of ${\mathbb {W}}_{i}$ is $-A_{i+1}\in {\mathbb {C}}^{n\times n}$ and the $(3+i,3+i)$ block of ${\mathbb {W}}_{i}$ is $-D_{i+1}\in {\mathbb {C}}^{p\times m}$ for $i=0:d_{A}-2$ and the (1, 1) block of ${\mathbb {W}}_{i}$ is $-A_{d_{A}-1}\in {\mathbb {C}}^{n\times n}$ and the $(d_{A}+1,d_{A}+1)$ block of ${\mathbb {W}}_{i}$ is $-D_{i+1}\in {\mathbb {C}}^{p\times m}$ for $i=d_{A}-1:d_{D}-2$. The rest of the diagonal blocks of ${\mathbb {W}}_i$ are square zero matrices, and more precisely, for $j = 0, 1, \ldots , i$ and $i= 0, 1, \ldots d_{{A}}-2,$ ${\mathbb {W}}_i(i+2-j, i+2-j) = 0_n$ and for $j = 0, 1, \ldots , i$ and $i= 0, 1, \ldots d_{{A}}-2,$

  $$\begin{aligned} {\mathbb {W}}_i(2i+4-j, 2i+4-j) = {\left\{ \begin{array}{ll} 0_p, &{} \text {if } \sigma \text { has a consecution at } j \\ 0_m, &{} \text {if } \sigma \text { has an inversion at } j \end{array}\right. }, \end{aligned}$$

  and for $j=0, 1, \ldots , i$ and $i=d_{A}-1:d_{D}-2$,

  $$\begin{aligned} {\mathbb {W}}_{i}(d_{A}+i+2-j,d_{A}+i+2-j)={\left\{ \begin{array}{ll} 0_p, \,\,\,\,\ {} &{}\text {if}\, \sigma \, \text {has a conseqution at}\, j \\ 0_m, &{}\text {if}\, \sigma \,\text {has an inversion at}\, j \end{array}\right. }. \end{aligned}$$

• (c) If $\sigma$ has a consecution at i, then the size of the zero block in $W_{21}$ block of ${\mathbb {W}}_i$ is $0_{p \times n}$ and if $\sigma$ has an inversion at i, then the size of the zero block in $W_{12}$ block of ${\mathbb {W}}_i$ is $0_{n \times m}$.

Proof

The proof is analogous to the proof of Lemma 3.4. $\square$

We have the following properties.

Lemma 6.2

Let $S(\lambda )$ be as in (1.5) with $A(\lambda ) = \sum \nolimits _{i=0}^{d_A}\lambda ^{i}A_i\in \mathbb C[\lambda ]^{n\times n}$, $D(\lambda ) = \sum \nolimits _{i=0}^{d_D}\lambda ^{i}D_{i}\in \mathbb C[\lambda ]^{p\times m}$ and suppose that $d_A< d_D$. Let $\sigma : \{0, 1, \ldots , d_{D}-1\} \rightarrow \{1, 2, \ldots , d_{D}\}$ be a bijection and consider Algorithms 5 and  6. Let $\{{\mathbb {N}}_0, {\mathbb {N}}_1, \ldots , {\mathbb {N}}_{d_{D}-2}\}$ and $\{{\mathbb {H}}_0, {\mathbb {H}}_1, \ldots , {\mathbb {H}}_{d_{D}-2}\}$ be given by Algorithms 5 and  6, respectively. Consider the sequence of block partitioned matrices $\{{\mathbb {W}}_i\}_{i=0}^{d_{D}-2}$ constructed by the Algorithm 4. Then we have the following.

• (a) For $0\le i \le d_{D}-2$, and $1 \le j\le i+2,$ the number of columns of ${\mathbb {N}}_i(:, j)$ is equal to the number of rows of ${\mathbb {W}}_i(j,:)$ so that the product ${\mathbb {N}}_i(:, j){\mathbb {W}}_i(j,:)$ is well defined.

• (b) For $0\le i \le d_{D}-2$, and $1 \le j\le i+2,$ the number of columns of ${\mathbb {W}}_i(:, j)$ is equal to the number of rows of ${\mathbb {H}}_i(j,:)$ so that the product ${\mathbb {W}}_i(:, j){\mathbb {H}}_i(j,:)$ is well defined.

• (c) For $i= 0:d_{A}-2$, the size of ${\mathbb {N}}_i$ is $\left[ \left( n + n c(\sigma (0:i)) + n \mathfrak {i}(\sigma (0:i)) \right) +\left( p + p c(\sigma (0:i)) + m \mathfrak {i}(\sigma (0:i))\right) \right] \times \left[ \left( n + n c(\sigma (0:i)) + n \mathfrak {i}(\sigma (0:i)) \right) +\left( p + p c(\sigma (0:i)) + m \mathfrak {i}(\sigma (0:i)) \right) \right]$ and for $i= d_{A}-1: d_{D}-2$ the size of ${\mathbb {N}}_i$ is

  $$\begin{aligned}{} & {} \left[ \left( n d_{A}\right) + \left( p+p c(\sigma (0:i)) + m \mathfrak {i}(\sigma (0:i))\right) \right] \\{} & {} \quad \times \left[ \left( n d_{A}\right) + \left( p + p c(\sigma (0:i)) + m \mathfrak {i}(\sigma (0:i))\right) \right] . \end{aligned}$$

• (d) For $i= 0:d_{A}-2$, the size of ${\mathbb {H}}_i$ is $\left[ \left( n + n c(\sigma (0:i)) + n \mathfrak {i}(\sigma (0:i))\right) + \left( m + p c(\sigma (0:i)) + m \mathfrak {i}(\sigma (0:i))\right) \right] \times \left[ \left( n + n c(\sigma (0:i)) + n \mathfrak {i}(\sigma (0:i)) \right) + \left( m + p c(\sigma (0:i)) + m \mathfrak {i}(\sigma (0:i)) \right) \right]$ and for $i= d_{A}-1: d_{D}-2$ the size of ${\mathbb {H}}_i$ is

  $$\begin{aligned}{} & {} \left[ \left( n d_{A}\right) + \left( m+p c(\sigma (0:i)) + m \mathfrak {i}(\sigma (0:i))\right) \right] \\{} & {} \quad \times \left[ \left( n d_{A}\right) + \left( m + p c(\sigma (0:i)) + m \mathfrak {i}(\sigma (0:i))\right) \right] . \end{aligned}$$

• (e) The matrix polynomials ${\mathbb {N}}_i$ and ${\mathbb {H}}_i$ are unimodular with $\det ({\mathbb {N}}_i) = \pm 1$ and $\det ({\mathbb {H}}_i) = \pm 1$.

Lemma 6.3

Let $S(\lambda )$ be an $(n+p) \times (n+m)$ RSMP as in (1.5) with $A(\lambda ) = \sum \nolimits _{i=0}^{d_A}\lambda ^{i}A_i\in \mathbb {C}[\lambda ]^{n\times n}$ and $D(\lambda ) = \sum \nolimits _{i=0}^{d_D}\lambda ^{i}D_{i} \in \mathbb {C}[\lambda ]^{p\times m}$ and $d_A < d_D$. Let $\sigma : \{0, 1, \ldots , d_{D}-1\} \rightarrow \{1, 2, \ldots , d_{D}\}$ be a bijection. Let $\{{\mathbb {W}}_i\}_{i=0}^{d_{D}-2}, \{{\mathbb {N}}_i\}_{i=0}^{d_{D}-2}$ and $\{{\mathbb {H}}_i\}_{i=0}^{d_{D}-2}$ be the sequences of block matrices constructed in Algorithms 4, 5, and 6, respectively. Also consider the numbers $\alpha _{i} = p c(\sigma (0:i)) + m \mathfrak {i}(\sigma (0:i))$, $\beta _{i} = p c(\sigma (i)) + m \mathfrak {i}(\sigma (i))$ and $\alpha _{i}' = n c(\sigma (0:i)) + n \mathfrak {i}(\sigma (0:i))$, $\beta _{i}' = n c(\sigma (i)) + n \mathfrak {i}(\sigma (i))$. Note that $\beta _{i} = p$ if $\sigma$ has a consecution at i and $\beta _{i} = m$, if $\sigma$ has an inversion at i, for $i = 0, 1, 2, \ldots , d_{D}-2$. Then we have the following.

• (a) For $1 \le i \le d_{D}-2$ we have

  $$\begin{aligned}{} & {} {\mathbb {N}}_i \left( \left[ \begin{array}{cc|cc} \lambda P_{d_{A}-i-2} &{} &{} &{} \\ &{} \lambda I_{\alpha _i'} &{} &{} \\ \hline &{} &{} \lambda Q_{d_{D}-i-2}&{} \\ &{} &{} &{} \lambda I_{\alpha _i} \\ \end{array} \right] -{\mathbb {W}}_i \right) {\mathbb {H}}_i\\{} & {} \qquad = \left[ \begin{array}{cc|cc} I_{\beta _i} &{} &{} 0 &{} 0 \\ &{} N_{i-1} \left( \left[ \begin{array}{cc} \lambda P_{d_{A}-i-1} &{} \\ &{} \lambda I_{\alpha '_{i-1}} \\ \end{array} \right] -W_{i-1} \right) H_{i-1} &{} 0 &{} -B \\ \hline 0 &{} 0 &{} I_{\beta _i} &{} \\ 0 &{} C &{} &{} N_{i-1}' \left( \left[ \begin{array}{cc} \lambda Q_{d_{D}-i-1} &{} \\ &{} \lambda I_{\alpha _{i-1}} \\ \end{array} \right] -W_{i-1}' \right) H_{i-1}' \\ \end{array} \right] \end{aligned}$$

• (b) For $i = 0$

  $$\begin{aligned} {\mathbb {N}}_0 \left( \left[ \begin{array}{cc|cc} \lambda P_{d_{A}-2} &{} &{} &{} \\ &{} \lambda I_{\alpha _0'} &{} &{} \\ \hline &{} &{} \lambda Q_{d_{D}-2} &{} \\ &{} &{} &{} \lambda I_{\alpha _0} \\ \end{array} \right] -{\mathbb {W}}_0 \right) {\mathbb {H}}_0 = \left[ \begin{array}{cc|cc} I_{\beta _0'} &{} &{} 0 &{} 0 \\ &{} A(\lambda ) &{} 0 &{} -B \\ \hline 0 &{} 0 &{} I_{\beta _0} &{} \\ 0 &{} C &{} &{} D(\lambda ) \\ \end{array} \right] \end{aligned}$$

Proof

The proof is analogous to the proof of Lemma 4.6. $\square$

Theorem 6.4

Let $S(\lambda )$ be an $(n+p) \times (n+m)$ system matrix given in (1.5) with $A(\lambda ) = \sum \nolimits _{i=0}^{d_A}\lambda ^{i}A_i\in \mathbb {C}[\lambda ]^{n\times n}$ and $D(\lambda ) = \sum \nolimits _{i=0}^{d_D}\lambda ^{i}D_{i} \in \mathbb {C}[\lambda ]^{p\times m}$ with $d_A < d_D$. Let $\sigma : \{0, 1, \ldots , d_{D}-1\} \rightarrow \{1, 2, \ldots , d_{D}\}$ be a bijection. Then any Fiedler pencil ${\mathbb {L}}_{\sigma }(\lambda )$ is a linearization for $S(\lambda )$ associated with any bijection $\sigma .$

Proof

The proof is analogous to the proof of Theorem 4.7. $\square$