# 4 Linear DAEs with Time-varying Coefficients

Theorem 4.1 Let $$E, A \in \mathbb C^{m,n}$$ and let $$$T,~Z,~T',~V \tag{4.1}$$$ be

Matrix as the basis of
$$T$$ $$\operatorname{kernel}E$$
$$Z$$ $$\operatorname{corange}E = \operatorname{kernel}E^H$$
$$T'$$ $$\operatorname{cokernel}E = \operatorname{range}E^H$$
$$V$$ $$\operatorname{corange}(Z^HAT)$$

then the quantities $$$r,~a,~s,~d,~u,~v \tag{4.2}$$$ defined as

Quantity Definition Name
$$r$$ $$\operatorname{rank}E$$ rank
$$a$$ $$\operatorname{rank}(Z^HAT)$$ algebraic part
$$s$$ $$\operatorname{rank}(V^HZ^HAT')$$ strangeness
$$d$$ $$r-s$$ differential part
$$u$$ $$n-r-a$$ undetermined variables
$$v$$ $$m-r-a-s$$ vanishing equations

are invariant under local equivalence transformations and $$(E, A)$$ is locally equivalent to the canonical form

$$$\left(\begin{bmatrix} I_s & 0 & 0 & 0 \\ 0 & I_d & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & I_a & 0 \\ I_s & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}\right), \tag{4.3}$$$ where all diagonal blocks are square, except maybe the last one.

Some remarks on the spaces and how the names are derived for the case $$E\dot x = Ax +f$$ with constant coefficients. The ideas are readily transferred to the case with time-varying coefficients.

Let $x(t) = Ty(t) + T'y'(t),$

where $$y$$ denotes the components of $$x$$ that evolve in the range of $$T$$ and $$y'$$ the respective complement. (Since $$[T|T']$$ is a basis of $$\mathbb C^{n}$$, there exist such $$y$$ and $$y'$$ that uniquely define $$x$$ and vice versa). With $$T$$ spanning $$\ker E$$ we find that

$E \dot x(t) = ET\dot y(t) + ET'\dot y'(t)$

so that the DAE basically reads

$ET'\dot y'(t) = ATy(t) + AT'y'(t)+f,$

i.e. the components of $$x$$ defined through $$y$$ are, effectively, not differentiated. With $$Z$$ containing exactly those $$v$$, for which $$v^HE=0$$, it follows that

$Z^HET'\dot y'(t) = 0 = Z^HATy(t) + Z^HAT'y'(t)+Z^Hf,$

or

$Z^HATy(t) = -Z^HAT'y'(t)-Z^Hf,$

so that $$\operatorname{rank}Z^HAT$$ indeed describes the number of purely algebraic equations and variables in the sense that it defines parts of $$y$$ (which is never going to be differentiated) in terms of algebraic relations (no time derivatives are involved).

With the same arguments and with $$V=\operatorname{corange}Z^HAT$$, it follows that

$V^HZ^HAT'y'(t) = -V^HZ^HATy(t) -V^HZ^Hf=-V^HZ^Hf,$

is the part of $$E\dot x = Ax + f$$ in which those components $$y'$$ that are also differentiated are algebraically equated to a right-hand side. This is the strangeness (rather in the sense of skewness) of DAEs that variables can be both differential and algebraic. Accordingly, $$\operatorname{rank}V^HZ^HAT'$$ describes the size of the skewness component.

Outlook: If there is no strangeness, the DAE is called strangeness-free. Strangeness can be eliminated through iterated differentiation and substitution. The needed number of such iterations (that is independent of the the size $$s$$ of the strange block here) will define the strangeness index.

Example 4.1 With basic scalings and state transforms, one finds for the coefficients of Example 1.2 that: $(E, A) \backsim \left( \begin{bmatrix} I_2 & 0 \\ 0 & 0 \end{bmatrix} , \begin{bmatrix} 0 & 0 \\ 0 & I_1 \end{bmatrix} \right).$

We compute the subspaces as defined in (4.1):

Matrix as the basis of/computed as
$$T=\begin{bmatrix} 0 \\ I_1 \end{bmatrix}$$ $$\operatorname{kernel}\begin{bmatrix} I_2 & 0 \\ 0 & 0 \end{bmatrix}$$
$$Z=\begin{bmatrix} 0 \\ I_1 \end{bmatrix}$$ $$\operatorname{corange}\begin{bmatrix} I_2 & 0 \\ 0 & 0 \end{bmatrix}=\operatorname{kernel}\begin{bmatrix} I_2 & 0 \\ 0 & 0 \end{bmatrix}^H$$
$$T'=\begin{bmatrix} I_2 \\ 0 \end{bmatrix}$$ $$\operatorname{cokernel}\begin{bmatrix} I_2 & 0 \\ 0 & 0 \end{bmatrix}=\operatorname{range}\begin{bmatrix} I_2 & 0 \\ 0 & 0 \end{bmatrix}^H$$
$$Z^HAT=I_1$$ $$\begin{bmatrix} 0 \\ I_1 \end{bmatrix}^H\begin{bmatrix} 0 & 0 \\ 0 & I_1 \end{bmatrix}\begin{bmatrix} 0 \\ I_1 \end{bmatrix}$$
$$V=0$$ $$\operatorname{corange}(Z^HAT) = \operatorname{kernel}I_1^H\phantom{\begin{bmatrix} 0 \\ I_1 \end{bmatrix}}$$
$$Z^HAT'=0_{2\times 1}$$ $$\begin{bmatrix} 0 \\ I_1 \end{bmatrix}^H\begin{bmatrix} 0 & 0 \\ 0 & I_1 \end{bmatrix}\begin{bmatrix} I_2 \\ 0 \end{bmatrix}$$

and derive the quantities as defined in (4.2):

Name Value Derived from
rank $$r=2$$ $$\operatorname{rank}E = \operatorname{rank}\begin{bmatrix} I_2 & 0 \\ 0 & 0 \end{bmatrix}$$
algebraic part $$a=1$$ $$\operatorname{rank}Z^HAT = \operatorname{rank}I_1$$
strangeness $$s=0$$ $$\operatorname{rank}V^HZ^HAT' = \operatorname{rank}0_{2\times 1}$$
differential part $$d=2$$ $$d=r-s=2-0$$
undetermined variables $$u=0$$ $$u=n-r-a=3-2-1$$
vanishing equations $$v=0$$ $$v=m-r-a-s=3-2-1-0$$

Example 4.2 With more involved scalings and state transforms, one finds for the coefficients of the linearized and spatially discretized Navier-Stokes equations (see Exercise I) that: $(\mathcal E, \mathcal A) = \left( \begin{bmatrix} M & 0 \\ 0 & 0 \end{bmatrix} , \begin{bmatrix} A & B^H \\ B & 0 \end{bmatrix} \right) \backsim \left( \begin{bmatrix} I_{n_1} & 0 & 0 \\ 0 & I_{n_2} & 0 \\ 0 & 0 & 0\end{bmatrix} , \begin{bmatrix} A_{11} & A_{12} & I_{n_1} \\ A_{21} & A_{22} & 0 \\ I_{n_1} & 0 & 0\end{bmatrix} \right).$

We compute the subspaces as defined in (4.1):

Matrix as the basis of/computed as
$$T=\begin{bmatrix} 0 \\ 0 \\I_{n_1} \end{bmatrix}$$ $$\operatorname{kernel}\begin{bmatrix} I_{n_1} & 0 & 0 \\ 0 & I_{n_2} & 0 \\ 0 & 0 & 0\end{bmatrix}$$
$$Z=\begin{bmatrix} 0 \\ 0 \\I_{n_1} \end{bmatrix}$$ $$\operatorname{corange}\begin{bmatrix} I_{n_1} & 0 & 0 \\ 0 & I_{n_2} & 0 \\ 0 & 0 & 0\end{bmatrix}$$
$$T'=\begin{bmatrix} I_{n_1} & 0 \\ 0 & I_{n_2} \\ 0 & 0 \end{bmatrix}$$ $$\operatorname{cokernel}\begin{bmatrix} I_{n_1} & 0 & 0 \\ 0 & I_{n_2} & 0 \\ 0 & 0 & 0\end{bmatrix}$$
$$Z^HAT=0_{n_1}$$ $$\begin{bmatrix} 0 \\ 0 \\I_{n_1} \end{bmatrix}^H\begin{bmatrix} A_{11} & A_{12} & I_{n_1} \\ A_{21} & A_{22} & 0 \\ I_{n_1} & 0 & 0\end{bmatrix}\begin{bmatrix} 0 \\ 0 \\I_{n_1} \end{bmatrix}$$
$$V=I_{n_1}$$ $$\operatorname{corange}(Z^HAT) = \operatorname{kernel}0_{n_1}^H\phantom{\begin{bmatrix} 0 \\ I_1 \end{bmatrix}}$$
$$Z^HAT'=\begin{bmatrix} I_{n_1} & 0_{n_1\times n_2}\end{bmatrix}$$ $$\begin{bmatrix} 0 \\ 0 \\I_{n_1} \end{bmatrix}^H\begin{bmatrix} A_{11} & A_{12} & I_{n_1} \\ A_{21} & A_{22} & 0 \\ I_{n_1} & 0 & 0\end{bmatrix}\begin{bmatrix} I_{n_1} & 0 \\ 0 & I_{n_2} \\ 0 & 0 \end{bmatrix}$$

and derive the quantities as defined in (4.2):

Name Value Derived from
rank $$r=n_1+n_2$$ $$\operatorname{rank}E = \operatorname{rank}\begin{bmatrix} I_{n_1} & 0 & 0 \\ 0 & I_{n_2} & 0 \\ 0 & 0 & 0\end{bmatrix}$$
algebraic part $$a=0$$ $$\operatorname{rank}Z^HAT = \operatorname{rank}0_{n_1}$$
strangeness $$s=n_1$$ $$\operatorname{rank}V^HZ^HAT' = \operatorname{rank}\begin{bmatrix} I_{n_1} & 0_{n_1\times n_2}\end{bmatrix}$$
differential part $$d=n_2$$ $$d=r-s=(n_1 + n_2) - n_1$$
undetermined variables $$u=n_1$$ $$u=n-r-a=(n_1+n_2+n_1)-(n_1+n_2)-0$$
vanishing equations $$v=0$$ $$v=m-r-a-s=(n_1+n_2+n_1)-(n_1+n_2)-n_1$$

Theorem 4.2 (see Kunkel/Mehrmann, Thm. 3.9) Let $$E\in \mathcal C^l(I, \mathbb C^{m,n})$$ with $$\operatorname{rank}E(t)=r$$ for all $$t\in I$$. Then there exist smooth and pointwise unitary (and, thus, nonsingular) matrix functions $$U$$ and $$V$$, such that

$U^HEV = \begin{bmatrix} \Sigma & 0 \\ 0 & 0 \end{bmatrix}$ with pointwise nonsingular $$\Sigma \in \mathcal C^l(I, \mathbb C^{r,r})$$.

Theorem 4.3 Let $$E, A \in \mathcal C^l(I, \mathbb C^{m,n})$$ be sufficiently smooth and suppose that $$$r(t) = r, \quad a(t)=a, \quad s(t)=s \tag{4.4}$$$

for the local characteristic values of $$(E(t), A(t))$$. Then $$(E, A)$$ is globally equivalent to the canonical form $$$\left( \begin{bmatrix} I_s & 0 & 0 & 0 \\ 0 & I_d & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & A_{12}& 0 & A_{14} \\ 0 & 0 & 0 & A_{24} \\ 0 & 0 & I_a & 0 \\ I_s & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \right ). \tag{4.5}$$$

All entries are again matrix functions on $$I$$ and the last block column in both matrix functions of (4.5) has size $$u=n-s-d-a$$.

Proof. In what follows, we will tacitly redefine the block matrix entries that appear after the global equivalence transformations. The first step is the continous SVD of $$E$$; see Theorem 4.3. \begin{align*} (E,A) & \sim \left(\begin{bmatrix} \Sigma & 0 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}\right) \\ %%%%%%%%%%%%%%%%%%%%%%%%%% & \sim \left(\begin{bmatrix} I_r & 0 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}\right) \\ %%%%%%%%%%%%%%%%%%%%%% & \sim \left(\begin{bmatrix} I_r & 0 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} A_{11} & A_{12}V_1 \\ U_1^HA_{21} & U_1^HA_{22}V_1 \end{bmatrix} - \begin{bmatrix} I_r & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \\ 0 & \dot V_1 \end{bmatrix} \right) \\ %%%%%%%%%%%%%%%%%%%%%% & \sim \left(\begin{bmatrix} I_r & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} A_{11} & A_{12} & A_{13}\\ A_{21} & I_a & 0 \\ A_{31} & 0 & 0 \end{bmatrix}\right) \\ %%%%%%%%%%%%%%%%%%%%%% & \sim \left(\begin{bmatrix} V_2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} A_{11}V_2 & A_{12} & A_{13}\\ A_{21}V_2 & I_a & 0 \\ U_2^HA_{31}V_2 & 0 & 0 \end{bmatrix} - \begin{bmatrix} \dot I_r & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \dot V_2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \right)\\ %%%%%%%%%%%%%%%%%%%%%% & \sim \left(\begin{bmatrix} V_{11} & V_{12} & 0 & 0 \\ V_{21} & V_{22} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} A_{11} & A_{12} & A_{13} & A_{14} \\ A_{21} & A_{22} & A_{23} & A_{24} \\ A_{31} & A_{32} & I_a & 0 \\ I_s & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \right)\\ %%%%%%%%%%%%%%%%%%%%%% & \sim \left(\begin{bmatrix} I_s & 0 & 0 & 0 \\ 0 & I_d & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & A_{12} & A_{13} & A_{14} \\ 0 & A_{22} & A_{23} & A_{24} \\ 0 & A_{32} & I_a & 0 \\ I_s & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} I_s & 0 & 0 & 0 \\ 0 & I_d & 0 & 0 \\ 0 & -A_{32} & I_a & 0 \\ I_s & 0 & 0 & I_a \end{bmatrix} \right) \\ %%%%%%%%%%%%%%%%% & \sim \left(\begin{bmatrix} I_s & 0 & 0 & 0 \\ 0 & I_d & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & A_{12} & A_{13} & A_{14} \\ 0 & A_{22} & A_{23} & A_{24} \\ 0 & 0 & I_a & 0 \\ I_s & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}\right)\\ %%%%%%%%%%%%%%%%% & \sim \left(\begin{bmatrix} I_s & 0 & 0 & 0 \\ 0 & I_d & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & A_{12} & 0 & A_{14} \\ 0 & A_{22} & 0 & A_{24} \\ 0 & 0 & I_a & 0 \\ I_s & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}\right)\\ %%%%%%%%%%%%%%%%% & \sim \left(\begin{bmatrix} I_s & 0 & 0 & 0 \\ 0 & Q_2 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & A_{12}Q_2 & 0 & A_{14} \\ 0 & A_{22}Q_2-\dot Q_2 & 0 & A_{24} \\ 0 & 0 & I_a & 0 \\ I_s & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}\right) \\ %%%%%%%%%%%%%%%%% & \sim \left(\begin{bmatrix} I_s & 0 & 0 & 0 \\ 0 & I_d & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & A_{12} & 0 & A_{14} \\ 0 & 0 & 0 & A_{24} \\ 0 & 0 & I_a & 0 \\ I_s & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}\right), \end{align*} where the final equivalence holds, if $$Q_2$$ is chosen as the (unique and pointwise invertible) solution of the linear matrix valued ODE $\dot Q_2 = A_{22}(t)Q_2 , \quad Q_2 (t_0 ) = I_d.$ Then, $$A_{22}$$ vanishes because of the special choice of $$Q_2$$ and $$E_{22}$$ becomes $$I_d$$ after scaling the second block line by $$Q_2^{-1}$$.