Canonical Subspaces of Linear Time-Varying Differential-Algebraic Equations and Their Usefulness for Formulating Accurate Initial Conditions

Accurate initial conditions have the task of precisely capturing and fixing the free integration constants of the flow considered. This is trivial for regular ordinary differential equations, but a complex problem for differential-algebraic equations (DAEs) because, for the latter, these free constants are hidden in the flow. We deal with linear time-varying DAEs and obtain an accurate initial condition by means of applying both a reduction technique and a projector based analysis. The highlighting of two canonical subspaces plays a special role. In order to be able to apply different DAE concepts simultaneously, we first show that the very different looking rank conditions on which the regularity notions of the different concepts (elimination of unknowns, reduction, dissection, strangeness, and tractability) are based are de facto consistent. This allows an understanding of regularity independent of the methods.


Introduction
In the context of generalized eigenvalue problems, regular matrix pencils, and descriptor systems, so-called spectral projectors and their associated subspaces receive special attention. In contrast, they have received little attention in the context of the analysis of differentialalgebraic equations (DAEs). We dedicate the present work to this question.
In doing so, we take as a basis the regularity notion that adopts the reduction idea presented in [24]. We will prove that this notion captures equally proper the very different approaches from [4,15,24,16,17]. In particular, we provide a set of characteristic values from which one can derive the characteristic values of the different approaches, and which, in this sense, are method-independent.
We deal both with DAEs in standard form, and DAEs with properly involved derivative term, with sufficiently smooth coefficient functions E, F : I → R m×m , and A : I → R m×k , D : I → R k×m , B : I → R m×m , respectively.
The properly involved derivative is merely useful when it comes to exact smoothness issues of the solutions. Such properties do not matter here, since we assume continuously differentiable solutions, as is widely customary and even necessary for the use of the related concepts. Here we are interested in the time-varying subspaces of R m where the solution values are located. The canonical subspaces S can , N can , as well as the associated projector function Π can should not depend on how the leading term of the DAE is written down.
We will prove that any regular DAE (1) or (2) with sufficiently smooth coefficients possesses a well-defined canonical projector function Π can with associated subspaces S can and N can in analogy to, and as a generalization of, the spectral projector. Then the reduction procedure from [24] provides a basis of the canonical subspace S can .
The knowledge of the subspace N can allows to elaborate accurately stated initial conditions in the sense of [18,Definition 2.3] and [17,Theorem 2.52], i.e., the accurate capture of the implicit free integration constants. Using a basis of the corresponding canonical subspace S * can to the adjoint DAE, we obtain a suitable matrix which allows to formulate accurate initial conditions.
Accurately stated initial conditions are something quite different from consistent initial values. The latter are needed in integration codes so far, but the generation of consistent initial data is a problem on its own, e.g., [5,3,2].
The least-squares collocation procedures for initial and boundary value problems with higher index differential-algebraic equations proposed and analyzed in [11,10,9,8,13,14,12] work incredibly good, but one has to provide accurate initial and boundary conditions. Moreover, for the time-stepping version [7] for systems with dynamic degrees of freedom, one needs also accurately stated transition conditions from one window to the next.
The paper is structured as follows: In Section 2, as an introduction, we recapitulate the state of affairs for matrix pencils and DAEs with constant coefficients as well as for semiexplicit index-one DAEs. Then, in Section 3, we use the reduction approach of [24] , agree on the regularity notion, and provide the first canonical subspace S can , which is related to the flow of the homogeneous DAE. The following Section 4 is concerned with a comparison of different concepts, which in the end will show the general validity of the agreed notion of regularity together with its characteristics. This allows the simultaneous use of the results of the different associated theories. In Section 5 we elaborate what is meant by accurate initial conditions and the role of the second canonical subspace N can . The practically most important result is then stated in Theorem 6.2 in Section 6. In this theorem, we derive a matrix, suitable for formulating accurate initial conditions, by using information coming from the adjoint DAE. Further, a nontrivial example is discussed to some extend in Section 7. There is an appendix with additional information about some technical details.

A review of two well-known cases for motivation, and orientation 2.1. A look back at the well-understood regular matrix pencils
Given the ordered pair {E, F} of matrices E, F ∈ R m×m , we consider the expression p(λ ) = det(λ E + F) being a polynomial in λ , and the so-called matrix pencil λ E + F. If the polynomial does not vanish identically, the matrix pencil is said to be regular. In turn, the pair {E, F} is said to be regular then.
For each regular pair {E, F}, there exist two integers d, µ ≥ 0, µ ≤ m − d, and nonsingular matrices L, K ∈ R m×m such that Here and in the following, I k ∈ R k×k denotes the identity matrix for k ∈ N. Thereby, N is absent if d = m. Otherwise, N ist nilpotent with index µ, i.e., N µ = 0, N µ−1 = 0. Both integers µ, d, the sizes of the blocks N and W , their eigenstructure as well as the subspaces are uniquely determined by the matrix pencil or the pair {E, F}. Formula (3) describes the Weierstraß-Kronecker form 1 of the matrix pencil and pair, respectively, and µ is the Kronecker index. We refer to [17,Proposition 1.3] for corresponding proofs, because there the proof is arranged in such a way that also the less often recognized fact (4) At this place it should be noted that, in the linear algebra context, for a regular pair {E, F} with d < m, so-called complementary deflating subspaces corresponding to its finite and infinite eigenvalues play their role. Π can is then the (right) spectral projector onto the right deflating subspace and L −1 I d 0 0 0 L is the (left) spectral projector onto the left deflating subspace. The latter hardly plays a role in DAE theory, which is why the addition "right" is omitted here for the spectral projector Π can . It is worth adding that always ker E ⊆ N can holds true.
The linear constant coefficient DAE associated with the regular pair {E, F}, decomposes corresondingly into In [1,26], the term Quasi-Weierstraß is used to emphasize that, unlike the standard Kronecker form, no Jordan form is required for N and W here. 2 In terms of the proof in [17,Page 6]: Each solution of the DAE (5) has the form Clearly, the integer d is at the same time the dynamical degree of freedom of the system (5). The case d = 0, and hence Π can = 0, S can = {0} may happen. At this place, we emphasize that, when dealing with DAEs, one has to assume that neither the transformation matrices K, L nor the canonical subspaces together with spectral projectors are practically available with little effort. Calculating them is considerably easier 4 than solving general eigenvalue problems, but still quite complex and rather hopeless for the case of variable coefficients, which we are actually interested in.
In case of the homogeneous DAE (5) with d > 0, the flow is completely restricted to remain in the subspace S can and in turn, at each time a ∈ I , and arbitrary x a ∈ S can , there is exactly one solution passing through, i.e., x(a) = x a . The situation is quite different for nonhomogeneous DAEs. Only if the function g vanishes identically, 5 and in turn v(t) ≡ 0, then the flow is again restricted to S can . The property g(t) ≡ 0 is practically very unlikely and unrecognizable. Otherwise, one is confronted with nontrivial parts v(t) in (6). In particular, for given a ∈ I , x a ∈ S can , it results that and, obviously, the conditions or, equivalently, Π can x(a) = x a , imply u(a) = u a , and vice versa.
Also for x a ∈ R m (instead of x a ∈ S can ) we have , and the conditions or, equivalently, Π can x(a) = Π can x a , imply u(a) = u a , and vice versa. We emphasize the unique role of the subspace N can in the DAE context shown here. Only the knowledge of this subspace allows an accurate fixation of the corresponding single solution with property (10), and this even without knowledge of the projector Π can itself.
Since each function value x(a) may incorporate a nontrivial v(a), the initial condition x(a) = x a ∈ R m includes the necessary condition v(a) = v a which means that the corresponding differentiations in (7) for providing v(a) must be carried out. To really calculate such a kind of consistent initial values is quite laborious and uncertain. However, if x a is not consistent, then the initial value problem (IVP) is not solvable.
On the other hand, if one assumes that the initial conditions are to fix a unique solution from the flow, then exactly d initial conditions shall be set. The following three equivalent conditions are useful, in which the matrix G a ∈ R d×m has full row-rank d and its nullspace coincides with N can , that is, ker G a = N can , and Π ∈ R m×m is any projector matrix such that ker Π = N can , among others Π can is allowed. Also, for any sufficiently smooth inhomogeneity q, an IVP with any of these conditions is uniquely solvable without any further consistency conditions related to q having to be satisfied. In this sense, these initial conditions are accurately formulated. We conclude our brief review by showing that various fundamental matrices can be defined for the DAE, all of which are now pointwise singular, have constant rank d, and whose pointwise image is exactly S can . Typical here are on the one hand the maximal-size version X (t) using the canonical projector, and on the other hand the minimal-size version X b using a basis C of S can given by Both versions have their justification. The canonical projector contains more information, namely that about both N can and S can , and the corresponding maximal fundamental solution has semigroup properties analogous to the ODE case, e.g., [17], which is very helpful for analytical studies. On the other hand, it is usually easier to compute a basis numerically than a projector, especially if it is not an orthoprojector and thus one needs, say, the bases of two subspaces. Among other things, we also show below a new way to compute the canonical projector Π can which seems to be more convenient than the iterative original version in [21]. In the present paper, by means of the reduction technique presented in Section 3, a basis C of S can and a basis C * of the subspace S * can associated with the adjoint matrix pair {−E * , F * } are determined. From the relations imC = S can , kerC * * E = N can which are valid according to Theorem 6.2, one can then derive the projector by usual numerical algebra techniques. Moreover, we may choose G a = C * * E for IVPs.

Semi-explicit index-one DAEs
Let the matrix function F : I → R m×m be sufficiently smooth. The pair given by is associated with the DAE Ex ′ + Fx = q; in more detail, Let the entry F 22 (t) ∈ R (m−r)×(m−r) remain nonsingular on I . Then the DAE is regular with index one and posesses the basic subspaces such that R m = S(t) ⊕ N.
For better clarity, we usually omit the t-argument in the following. The relations are then meant pointwise.
Each solution x : I → R m of the DAE has the form in which U denotes the solution of x 1 (a) can be predefined arbitrarely by initial conditions, and Hereby d = r is the dynamical degree of freedom. Further, if q = 0, in turn f = 0, the solutions simplify to Since C(a) is a basis of S(a), and if S can (a) is to be again the subspace containing all solution values of the homogeneous DAE at time a, it results that S = S can . Next we are looking for an appropriate complementary to S can subspace N can in view of accurately stated initial conditions. The sought subspace must have dimension m − r.
x a1 x a2 ∈ R m with x a1 ∈ R r . Obviously, the condition x(a) − x a ∈ N implies x 1 (a) = x a 1 , and vice versa. Owing to the property (14), the subspace N is complementary to S(t) for all t. Here are the projector matrices P S (t) onto S(t) along N and P onto N ⊥ along N: , P = I r 0 0 0 .

All IVPs
are uniquely solvable, and each solution satisfies x 1 (a) = x a 1 , the subspace N plays the same role as N can in Subsection 2.1, so that N can = N, Π can = P S .
We underline the uniqueness of N can , which means that the condition x(a) − x a ∈Ñ with any (m − r)-dimensional subspaceÑ other than N involves terms x 2 (a) − x a 2 , thus parts of q(a). In particular, the condition x(a) − x a ∈ S(a) ⊥ , or equivalently C(a) * (x(a) − x a ) = 0, This will be an accurate initial condition, only if F 21 = 0, but then S(a) ⊥ coincides with N.
It is evident that accurate initial conditions should precisely determine the value x 1 (a), what can be done by one of the conditions with a matrix G a such that ker G a = N can , rank G a = d. For arbitrary given x a ∈ R m , each of these condition yields x 1 (a) = x a 1 . And this underlines the role of the canonical subspace N = N can . Remark 2.1. Like [24], most work on DAEs assumes continuously differentiable solutions. We will do the same in the present paper in order to include all relevant approaches, in particular the one of [24] which is especially easy to understand. This means that we must assume somewhat higher smoothness of the data than in those concepts like [17] where sharp solvability statements with the lowest possible smoothness properties are required. The difference is already visible in the solution representation (15): To have x 2 not only continuous but continuously differentiable, F −1 22 F 21 and F −1 22 q must be continuously differentiable. In concepts with properly involved derivative, one also accepts solution components x 2 being only contimuous, thus merely continuous data.
We point out that the matrix C(t) serves as a basis of the subspace S(t).
The assumption of a continuously differentiable solution goes along with the assumption that S is not only a continuous but also a continuously differentiable subspace varying in R m . Such smooth bases form the starting point on each level of the reduction procedure in [24].
Taking into account the possible lower smoothnesses in the reduction steps, for example as in [15], requires enormous technical effort that is hiding the essential principles.

Regular time-varying DAEs and their canonical subspace S can
We turn to the ordered pair {E, F} of matrix functions E, F : I → R m×m being sufficiently smooth, at least continuous, and consider the associated DAE as well as the accompanying time-varying subspaces in R m , In accordance with Section 2 we denote the subspace containing the flow of the homogeneous DAE at timet by S can (t), that is, the set of all possible function values x(t) of solutions of the DAE Ex ′ + Fx = 0, S can (t) := {x ∈ R m : there is a solution x : (t − δ ,t + δ ) ∩ I → R m of the homogeneous DAE such that x(t) =x},t ∈ I . 6 With an orthonormal basis C for S and the ansatz x = Cz + (I − CC * )x one can obtain a regular ODE for z together with uniquely solvable IVPs [22,Remarr 5.2]. The equation declared as essential underlying implicit ODE in [20], for example, is based on such an approach. Then, only as far as homogeneous DAEs are concerned, the initial condition C(a) * (x(a) − x a ) = 0 makes sense.
such that each IVP, witht ∈ I ,x ∈ R m , and sufficiently smooth q, is uniquely solvable without any consistency conditions for q or its derivatives at the pointt. This subspace will be further specified in Section 5 below.
In the present section we will agree on what regular DAEs are, and show that then the time-varying subspace S can (t) is well-defined on all I , and has constant dimension.
In Section 5 we will see that in case of a regular DAE both canonical subspaces are well-defined with dimensions independent oft ∈ I . The associated projector function thus becomes a generalization of the spectral projector for regular matrix pencils in Subsection 2.1.
with integers 0 ≤ r ≤ m and θ ≥ 0. Additionally, if θ = 0 and r < m, then the DAE is called regular with index one, but if θ = 0 and r = m, then the DAE is called regular with index zero.
We underline that any pre-regular pair {E, F} features three subspaces S(t), N(t), and N(t) ∩ S(t) having constant dimensions r, m − r, and θ , respectively.
We emphasize and keep in mind that now not only the coefficients are time dependent, but also the resulting subspaces. Nevertheless, we suppress in the following mostly the argument t, for the sake of better readable formulas. The equations and relations are then meant pointwise for all arguments.
The different cases for θ = 0 are well-understood. A regular index-zero DAE is actually a regular implicit ODE and S can = S = R m , N = {0}. Regular index-one DAEs feature S can = S, N can = N, e.g., [6,17], also Section 2.2. Note that r = 0 leads to S can = {0}. All these cases are only interesting here as intermediate results.
We turn back to the general case and describe the flow-subspace S can , and end up with a regularity notion associated with a regular flow.
The pair {E, F} is supposed to be pre-regular The first step of the reduction procedure from [24] is then well-defined, we refer to [24,Section 12] for the substantiating arguments.
Here we apply this procedure to homogeneous DAEs only.
We start by E 0 = E, F 0 = F, m 0 = m, r 0 = r, θ 0 = θ , and consider the homogeneous DAE By means of a basis Z 0 : From im[E 0 , F 0 ] = R m we derive that rank Z * 0 F = m 0 − r 0 , and hence the subspace S 0 = ker Z * 0 F has dimension r 0 . Obviously, each solution of the homogeneous DAE must stay in the subspace S 0 . Choosing a continuously differentiable basis C 0 : I → R m 0 ×r 0 of S 0 , each solution of the DAE can be represented as x = C 0 x [1] , with a function x [1] : I → R r 0 satisfying the reduced to size m 1 = r 0 DAE The pre-regularity assures that E 1 has constant rank r 1 = r 0 − θ 0 ≤ r 0 . Namely, we have Next we repeat the reduction step supposing that the new pair is pre-regular again, and so on. This yields m ≥ r 0 ≥ · · · ≥ r j ≥ r j−1 ≥ · · · ≥ 0. Denote by µ the smallest integer such that either r µ−1 = r µ > 0 or r µ−1 = 0. Then, it follows that ker E µ−1 ∩ S µ−1 = {0}, which means in turn that On the other hand, if r µ > 0 then E µ remains nonsingular and Otherwise, if r µ = 0, there is only the identically vanishing solution of the homogeneous DAE, x = 0. Moreover, for eacht ∈ I and each z ∈ imC(t), there is exactly one solution of the original homogeneous DAE passing through, x(t) = z.
As proved in [24], the ranks r = r 0 > r 1 > · · · > r µ−1 are independent of the special choice of the involved basis functions.
The property of pre-regularity does not necessarily carry over to the subsequent pair, as Example 3.2 shows.
is pre-gegular with m = 2, r = 1 and θ = 1. Choosing bases the subsequent pair is given by fails to have full row-rank m 1 , thus the pair {E 1 , F 1 } fails to be pre-regular. The associated to {E, F} homogeneous DAE possess the solutions in which α stands for an arbitrary smooth real function, which does not fit our idea of regularity.
The index µ and the ranks r = r 0 > r 1 > · · · > r µ−1 = r µ are called characteristic values of the pair and the DAE, respectively. By construction, for a regular pair it follows that r i+1 = r i − θ i , i = 0, · · · , µ − 1. Therefore, in place of the above µ + 1 rank values r 0 , . . . , r µ , the following rank and dimensions, can serve as characteristic quantities. Later it will become clear that these data also play a supporting role in other concepts, too. Remark 3.4. A predecessor version of the reduction procedure in [24] was already proposed and analyzed in [4] under the name elimination of the unknowns, even for more general pairs of rectangular matrix functions. There, an additional scaling of the respective pairs is incorporated to put them in partitioned form on each stage, cf. Appendix A.3, but this makes the description less clear. The regularity notion given in [4] is consistent with Definition 3.3. Another very related such reduction technique has been presented and extended a few years ago under the name dissection concept [15], see also Appendix A.3. This notion of regularity also agrees with Definition 3.3. As we shall see below, the regularity notions related to the strangeness-index concept and the tractability-index framework are consistent with Definition 3.3, too. Furthermore, the understanding of regular points e.g. in [25, Section 2.2.7] fits to this then also. Theorem 3.5. If the DAE (19) is regular on I with index µ and characteristics or, equivalently, (22), then S can (t) has dimension d = r − ∑ µ−2 j=0 θ j = r µ−1 for all t ∈ I , and the matrix function C : I → R m×d , C = C 0 · · ·C µ−2 , generated by the reduction procedure is a basis of S can .
we arrive at the sequence of subspaces living all in R m , showing dimensions r = r 0 > r 1 > · · · > r µ−1 , respectively. 8 Regarding that , and also the sequence of inclusions These inclusions seem to indicate a certain relationship to the approach in [5].
It is important to mention that pre-regularity and regularity persist and the characteristic values are invariant under equivalence transformations. An equivalence transformation of {E, F} is given by matrix functions L : I → R m×m being continuous and pointwise nonsingular and K : I → R m×m being continuously differentiable and pointwise nonsingular yielding the pairẼ,F,Ẽ For the proof concerning regularity we refer to [24].

On further regularity notions and their relations
We are concerned here with the regularity notions and approaches from [4,15,16,17] associated with the elimination procedure, the dissection concept, the strangeness reduction, and the tractability framework compared to Definition 3.3. The approaches in [4,15,16] are de facto special solution methods including reduction steps by elimination of variables and differentiations of certain variables. In contrast, the concept in [17] aims at a structural projector-based decomposition of the given DAE in order to analyze them subsequently.
We have already mentioned in Remark 3.4 that the elimination procedure in [4] is an earlier, but less elegant version of the reduction procedure in [24], which we have adopted in Section 3. The regularity definition [4, p. 58] agrees with Definition 3.3 in the matter and also with the name. However, it does not yet specify any characteristic values, which is why we will only refer to [24] in the remainder of this paper.
Each of the concepts is associated with a sequence of pairs of matrix functions, each supported by certain rank conditions that look very different. Thus also the regularity notions, which require in each case that the sequences are well-defined with well-defined termination, look very different. At the end of this section, we will know that all these regularity terms agree with our Definition 3.3, and that the characteristics (22) capture all the rank conditions involved.
When describing the different methods, traditionally the same terms are used, for example {E j , F j } for the matrix function pairs and r j for the ranks. However, they have different meanings in each instance. To avoid confusion, we label the different characters with corresponding exponents R (reduction), S (strangeness), D (dissection), and T (tractability), respectively, when there is a risk of confusion.
We first relate the regularity notion given by Definition 3.3 to the strangeness reduction concept.
Within the strangeness reduction framework the following five rank-values of the matrix function pair {E, F} play their role, e.g., [16, p. 59]: whereby T, T c , Z,V represent orthonormal bases (ONBs) of ker E, (ker E) ⊥ , (im E) ⊥ , and (im Z * FT ) ⊥ , respectively. The strangeness concept is tied to the requirement that r, a, and s are well-defined constant integers.
Proof. Let {E, F} be pre-regular, and T, T c be smooth ONBs what is possible owing to the constant rank r of E. Let Z,V be pointwise ONBs and N, S be given by (20). Then it holds S = ker Z * F by construction, and Z * FT has size (m − r) × (m − r). From ker Z * FT = T * (N ∩ S) we derive dim ker Z * FT = θ and hence Next we investigate the strangeness value s. For this aim we decompose Since T c * C X y = 0 means C X y ∈ ker E = N, thus C X y = 0, and therefore y = 0, we learn that rank T c * C X = r − θ .
Regardind the decomposition we conclude rankV = θ . Now we inspect the nullspace of V * Z * FT c . If z ∈ R r and V * Z * FT c z = 0, thus Z * FT c z ∈ im Z * FT , then there is a w ∈ R m−r such that Z * FT c z = −Z * FTw, further This yields T c z + Tw = ξ +C X v with a ξ ∈ N ∩ S and a v ∈ R r−θ . Taking into account that T and T c are ONBs of N and N ⊥ , respectively, we conclude z = T c * T c z = T c * C X v, and hence im T c * C X ⊆ kerV * Z * FT c .
On the other hand, choosing z ∈ im T c * C X we compute For better clarity of the following we add a further permutation transformation. Using the permutation matrix we arrive atẼ which is, of course, again equivalent to the original pair {E, F}. It remains checking its pre-regularity. The condition im[ẼF] = R m is evident andẼ has constant rank r = s + d. FromÑ we know that θ = s, such that the pair {Ẽ,F}, and in turn the original pair {E, F}, is preregular.
Assuming that the pair {E, F} is pre-regular, in turn the transformed pair {Ẽ,F} given by (30) is also pre-regular, we provide the successor pairs {Ẽ R 1 ,F R 1 } according to the reduction procedure in Section 3 and {E S 1 , F S 1 } according to the strangeness framework [16]. Using the basis functionsỸ andC, we form the reduced pair {Ẽ R 1 ,F R 1 } to {Ẽ,F}in accordance with Section 3, that is, The matrix functionẼ R 1 has size r × r, r = s + d and constant rank r [1] = d. If we look at the subspaces, it becomes clear that the reduced pair {Ẽ R 1 ,F R 1 } is pre-regular again, if and only if the following two conditions, are satisfied. In particular, (33) requires that the s × s matrix functionF 14 shows constant rank s − θ 1 = θ − θ 1 .
On the other hand, the matrix function pair {E S 1 , F S 1 } following the original pair {E, F} within the strangeness framework is given by replacing the entry (Ẽ) 14 in (30) by a zero matrix, which leads to We determine the strangeness characteristics of this pair. For this aim we use the corresponding bases and obtain and further r S 1 = d = r −θ = r [1] and a 1 = rank Z * 1 F S 1 T 1 = a+s+rankF 14 . Set r F14 = rank F 14 . LetṼ 14 denote an ONB of (imF 14 ) ⊥ .Ṽ 14 has size s × (s − r F14 ). Then, the matrix function forms an ONB of (im Z * 1 F S 1 T 1 ) ⊥ . Next we compute This results in the fact that the pair {E S 1 , F S 1 } is pre-regular, exactly if r F14 is constant, in turn a 1 is constant, and further u 1 = 0, so that s 1 = s − r F14 = dim kerF 14 . This requires exactly the conditions (32) and (33). In the consequence, the pairs {Ẽ R 1 ,F R 1 } and {E S 1 , F S 1 } are preregular simultaneously. If they are pre-regular, then s 1 = θ 1 , a 1 = m − r + s − s 1 = m − r + θ − θ 1 .
Then the following statements are true: (a) The strangeness index µ S is well-defined for {E, F}, and µ S = µ − 1. The associated characteristics are 9 Proof. We emphasize again that all characteristics involved here do not change under equivalence transformations [24,16,17,15]. (a): We perform the reduction process of the strangeness concept step by step and compare each level with the reduction from [24]. With the help of Lemmata 4.1 and 4.2 we get step by step the assertion. The reduction pairs in [24], which have lower dimension, turn out to be (to equivalence exactly) those parts of the pairs from [16] which still play a role for the further.  (22). Then the matrix pencil λ E + F is regular and, for i = 0, . . . , µ − 2, the quantity θ i is the number of Jordan blocks of order ≥ 2 + i within the nilpotent matrix N in the Weierstraß-Kronecker form (3).

Corollary 4.4. Let the pair {E, F} of matrices E, F ∈ R m×m be regular in the sense of Definition 3.3 with characteristic values
Proof. A constant matrix pair and the associated matrix pencil are known to be regular with Kronecker index µ exactly if it is regular with tractability index µ, e.g., [17,Chapter 1], and if so, l i = m − r T i−1 is the number of Jordan blocks of order ≥ i in the nilpotent matrix N in (3), i = 0, . . . , µ − 2. Owing to Theorem 4.3 we find that Remark 4.5. It seems to us very worth highlighting, that the θ -characteristics (22) make sense in all approaches and allow to determine all further characteristic values. Additionally, Corollary 4.4 reveals a feature independent of any method. Furthermore, so far it is clear that θ i coincides with s i and it is the dimension of intersecting subspaces in different approaches 10 : Remark 4.6. We have compared here the application of the reduction procedures from [16,15] and [24] to homogeneous DAEs, only. If, on the other hand, inhomogeneities q shall be considered, differentiations with respect to q must be made at each level. In [24], the resulting explicit relations are considered as being finished and, thus, neglected. So only the system of lower dimension, which is still of interest, is treated further. In contrast, in [16], all equations are further carried along. Apart from equivalence transformations at each level, this makes all the difference.

Accurate initial condition and the second canonical subspace N can
We emphasize again that we are not looking for consistent initial values here, but for an adequate formulation of initial conditions that lead to uniquely solvable IVPs, i.e., that precisely determine the free parameters of the flow of the DAE. We adopt the notion of accurately stated boundary condition [18, Definition 2.3] for this purpose. Consider the IVP are uniquely solvable and their solutions satisfy, on a compact interval I a ⊆ I , the inequality max t∈I a It should be remembered that regular higher-index DAEs lead to ill-posed problems, even if the initial conditions are stated accurately, e.g., [23].
Regular index-1 DAEs are studied in detail in [6,17], cf. also Subsection 2.2. Their dynamical degree of freedom is d = r = rank E(t), and one has simply N can = N, S can = S with N and S from (20). The related canonical projector function Π can : I → R m×m is given and also the solvability statements for IVPs with the initial condition x a ∈ N(a) = N can (a), x a ∈ R m , are proved. We underline that here x a is arbitrary and it is not necessarily a consistent value. On the other hand, Π can (a)x(a) = Π can (a)x a is always valid for the solution, while x(a) = x a cannot be expected in general.
While in the index-1 case one has simply N can = N and S can = S, in the case of regular higher-index pairs the canonical subspaces will be subspaces with higher and lower dimension, respectively, and with a constant K depending on the pair {E, F} and the interval I a only.
The assertions justify the notation N can .
Proof. Owing to Theorem 4.3 the pair {E, F} is regular with tractability index µ and characteristics This means in the projector-based framework that there is an admissible matrix function sequence (see [17], also Appendix A.4 below). The related nullspaces N i have dimensions θ i−1 , i = 1, · · · , µ, and dim N 0 = m − r, and The subspace N can is shown to be independent of the special choice of the admissible projector functions which form the projector function Π µ−1 , [17, Theorem 2.8]. Since we suppose sufficiently smooth E and F, a so-called fine decoupling sequence can be constructed starting with an arbitrary projector function Π 0 onto N 0 = ker E. Then, for Π µ−1 associated with a fine decoupling sequence, there is a further special projector function Q * 0 onto N 0 , such that (cf. [ Proof. Since γ, ∆γ ∈ im G a , we may choose x a , ∆x a ∈ R m such that γ = G a x a , ∆γ = G a ∆x a , and the IVPs (35) are uniquely solvable by Theorem 5.1(b). Furthermore, z := x− x * satisfies the homogeneous DAE and the initial condition G a z(a) − G a ∆x a = 0, and (37) implies max t∈I a what was to show.
Finally in this part, let us add that the maximal fundamental solution matrix of a regular DAE normalized at a ∈ I , that is the solution of the IVP, feature semigroup properties and im X (t, a) = S can (t), ker X (t, a) = N can (a), what we could see so similarly also in the case of constant matrix pairs in Section 2.1.

An useful representations of the matrix G a for accurately stated initial conditions and the projector function Π can
In the framework of the projector-based analysis [17] admissible matrix function sequences and incorporated admissible projector functions play their role, see also Appendix A.4. Owing to [17,Theorem 2.8] the subspaces N 0 + N 1 + · · · + N i do not depend of the special choice of the involved projector functions. For a regular DAE with index µ, it holds that with Π µ−1 given by an arbitrary admissible matrix function sequence. However, the possibilities of practical calculation are still limited, so we are looking for another way.
The homogeneous adjoint DAEs to the above DAEs (1) and (2), that is, Owing to [19, Theorem 3 and Corollary 2], the original DAE and its adjoint are regular with index µ at the same time, and they share the related characteristics, in particular the dynamical degree of freedom d. Let Π * can denote the canonical projector function associated with the adjoint DAE and let C * : I → R m×d be a basis of S * can = im Π * can , which can be provided, for example, using the procedure from Section 3.
Let's take a brief look at the semi-explisit index-1 DAE in Subsection 2.2 and consider its adjoint DAE. Example 6.1. The pair {E * , F * } below describes the DAE adjoint to the DAE in 2.2, , with F 22 remaining nonsingular.
The pair is regular with index one, with such that R m = S * ⊕ N * . The canonical projector function Π * can onto S * along N * and the orthogonal projector P * onto N ⊥ * along N * are given by Observe that evidently here d = r and and hence, for stating accurate initial conditions one can choose However, in the index-1 case in contrast to all higher-index cases there is the simpler possibility for the given DAE, namely, by choosing x(a) − x a ∈ ker E = N = N can . Theorem 6.2. Let the pair {E, F} be regular with index µ and canonical subspaces S can and N can , with bases C S can and C N can , respectively. Then the adjoint pair {−E * , F * − E * ′ } is also regular with the same characteristics. Moreover, with bases C S * can , C N * can of their canonical subspaces S * can and N * can , it results that Proof. We emphasize again that with the smoothness assumed here in general, it does not matter whether one chooses a homogeneous DAE in standard form, Ex ′ + Fx = 0, or a DAE with proper involved derivative, A(Dx) ′ +Bx = 0, to a given regular pair {E, F}, and E = AD, B = F A D ′ , see Appendix A.1. The pleasant symmetry of the DAE with proper involved derivative to its adjoint in many cases facilitates the investigation, although −D * (A * y) ′ + B * y = 0 is only another notation for −E * y ′ + (F * − E * ′ )y = 0. By [19,Theorem 3], the adjoint pair inherits the regularity along with characteristics from the regular pair {E, F}.
Owing to [19,Lemma 3] we know that in which D − and A * − are special generalized inverses corresponding to so-called complete decouplings. Regarding the relations ker DΠ can = ker Π can , im Π * can A * − = im Π * can , we derive On the other hand, writing shorter C * = C S * can it holds that and hence ker Π can = kerC * * AD. The second relation in the assertion of the theorem is valid for symmetry arguments.
Having the basis C =: C S can of S can , and a basis C N can of kerC * * AD = N can , respectively, the m × m matrix function M := [C S can C N can ] remains nonsingular everywhere on I , and the canonical projector is given by

We investigate the linear DAE E(t)x ′ (t) + F(t)x(t) = q(t) given by its coefficients
whereby the resulting blocks of different sizes with only zero entries are all denoted by 0.
Emphasizing the block structure we may write the homogeneous DAE as  Swapping the first two lines and also the variables leads to  what shows Hessenberg structure of size three. The DAE (41) is regular with index µ = 3 and features the sizes m = 7, m 1 = 3, m 2 = 3, m 3 = 1, dynamical degree of freedom d = 4, as well as characteristical values r T 0 = r T 1 = r T 2 = 6, r T 3 = 7. 11 In terms of Definition 3.3 and (22) one has r = 6, θ 0 = θ 1 = 1, and θ 2 = 0.
We choose a smooth basis C B of ker B * such that im B = im Ω, rank Ω = 1.
It is evident that By calculating the general solution of the DAE (41) and inspecting the solution structure one obtains that

An admissible matrix function sequence for DAE (41)
We start by and further Next we derive Regarding that im G 0 = im G 1 = im G 2 and owing to [17,Proposition 3.20] we determine the rank of G 3 = G 2 + B 2 Q 2 without knowing B 2 in detail: The projector function 12 Π 2 is not identical with the canonical projector function Π can since its image does not coincide with S can . The determination of Π can according to [17] is much more complex. But owing to [17,Theorem 2.8] which lists invariances of the construction of admissible sequences, it holds that 7.2. Accurately stated initial conditions to (41) by using Π 2 The initial condition to (41), with a matrix G a ∈ R d×m , is accurately stated, if im G a = R d , ker G a = ker Π can .
We now intend to build a suitable matrix G a . More precisely, we aim for a matrix function G : I → R d×m , such that G(a) may serve as G a for each arbitrary a ∈ I . Introducing a matrix function H : 12 We call attention to an error in the projector representation in [7,Section 6]. There the term (I 3 − Ω)AΩ is missing. Instead of the above correct Π 2 there the incorrect version  is given.

Below we choose 13
In particular, for a = 0, this yields and start the reduction of the pair of matrix functions featuring size 7 × 7, rank E 0 = 6, by choosing and forming This leads to the following basis C 0 of the subspace S 0 , and also to the reduced pair (size 6 × 6), 13 Note that H = V * can be chosen, with any basis V of ker B * .
Remember that C B is a smooth basis of ker B * (42) and Ω is the orthoprojector onto im B. With we arrive at the smooth basis C 1 to the subspace S 1 , as well as the next reduced pair, (size 5 × 5), we obtain as well as the basis Finally, taking the next matrix function E 3 (size 4 × 4) remains nonsingular, namely , and, hence the matrix function C = C 0 C 1 C 2 showing size 7 × 4, serves as basis of S * can 14 . We are mainly interested in the matrix function and its three-dimensional nullspace R is then called border-projector function.
Using any proper factorization of the leading coefficient E, the standard form DAE Ex ′ + Fx = q can be rewritten with B = F − AD ′ as DAE with properly stated leading term or DAE with properly involved derivative, A(Dx) ′ + Bx = q..
Of course, on the other hand, starting from a DAE with properly involved derivative, A(Dx) ′ + Bx = q, one immediately gains the DAE in standard form A properly involved derivative is essential when rigorous solvability statements are required and only the component Dx can be expected to be continuously differentiable. In the case of DAEs with proper involved derivative, the DAE and its adjoint show a formal symmetry that proves beneficial for the analysis. However, if, as here, we are concerned with the description of the time-varying subspaces in R m that capture the solution values, then the form of the inclusion of the derivative does not matter, in particular, and both DAE forms share their canonical projector Π can and the related canonical subspaces S can and N can . (2) We sketch here a modification of the reduction procedure from [24] for DAEs with properly involved derivative (2).

A.2. Modification of the reduction procedure for DAEs
We start by A 0 = A, D 0 = D, B 0 = B, m 0 = m, r 0 = r and consider the homogeneous DAE By means of a basis Z 0 : I → R (m 0 −r 0 )×m 0 of (im A 0 ) ⊥ = ker A * 0 and a basis Y 0 : I → R r 0 ×m 0 of im A 0 we divide the DAE into the two parts From im[A 0 D 0 B 0 ] = R m we derive that rank Z * 0 B = m 0 − r 0 , and hence. the subspace S 0 = ker Z * 0 B = ker Z * 0 F has dimension r 0 . Each solution of the homogeneous DAE must stay in the subspace S 0 . Choosing a basis C 0 : I → R r 0 ×m 0 of S 0 , each solution of the DAE can be represented as x = C 0 x [1] , with a function x [1] : I → R r 0 satisfying the reduced to size m 1 = r 0 DAE given below. In contrast to [24] where the basis C 0 is required to be continuously differentiable, we suppose now a continuous basis C 0 which has a continuosly differentiable part D 0 C 0 . Using a pointwise generalized inverse (D 0 C 0 ) − of D 0 C 0 we may write x [1] , [1] .
This leads to a DAE living in R m 1 , m 1 := r 0 , with properly involved derivative, x [1] = 0.

A.3. Basic steps by Cistyakov and Jansen
Let the pair {E, F}, E, F : I → R m×m , be pre-regular with constants r and θ according to Definition 3.1. We take over some notations from Section 4. Let T, T c , Z, and Y represent bases of ker E, (ker E) ⊥ , (im E) ⊥ , and im E, respectively. The matrix function Z * FT has size (m − r) × (m − r), rank m − r − θ = a, and ker Z * FT = T + (N ∩ S) has dimension θ . By scaling with [Y Z] * one splits the DAE Owing to the pre-regularity, the (m − r) × m matrix function Z * F features full row-rank m − r. We keep in mind that S = ker Z * F has dimension r.

A.3.1. Elimination by Cistyakov
Taking a nonsingular matrix function K of size m × m such that Z * FK =: [F 21F22 ], withF 22 being nonsingular, the transformation x = Kx turns (49) into Next one eliminates the variablex 2 in the transformed version of (48), which yields a DAE forx 1 =: x [1] , A further look at the matter shows that we are dealing with a special basis of the subspace S, namely whereby this early predecessor of the method described in [24] can now be classified as its special version. Note that the procedure in [24] allows for the choice of an arbitrary basis for S.
It should be further mentioned, that in [4] the elimination method is applied not only for pre-regular pairs but for general rectangular matrix functions E, F.

A.3.2. Dissectionn by Jansen
The approach in [15] needs several more splittings. As before let T, T c , Z, and Y represent bases of ker E, (ker E) ⊥ , (im E) ⊥ , and im E, respectively. Additionally, let V,W be bases of (im Z * FT ) ⊥ , and im Z * FT . By construction, V has size (m − r) × a and W has size (m − r) × θ . One starts with the transformation The background is the associated possibility to suppress the derivative of the nullspace-part Tx n similarly as in the context of properly formulated DAEs and to set Ex ′ = ET cx′ 1 + ET c′x 1 + ET ′x 2 , which, however, does not play a role here where altogether continuously differentiable solutions are assumed. Furthermore, an additional partition of the derivativefree equation (49) by means of the scaling with [V W ] * is applied, which results in the system The matrix function W * Z * FT c has full row-rank θ and V * Z * FT has full row-rank a. Now comes another split. Choosing bases G, H of kerW * Z * FT c ⊂ R θ and kerV * Z * FT ⊂ R a , as well as bases of respective complementary subspaces, we transform x 2 = H c H x 2x2 = x 2,1 x 2,2 ,x 2 = H cx 2,1 + Hx 2,2 .
Overall, therefore, the latter procedure presents again a transformation, namely and we realize that we have found again a basis of the subspace S, namely which makes the dissection approach a special case of [24]. The characteristic values together with the index are formally adapted to the values of the tractability index. It starts with r D 0 = r, and is continued in ascending order with r D i+1 = r D i + a i = r D i + rank Z + i F i T i etc. until r D µ−1 < r D µ = m, [15,Definition 4.13]. It should be noted, however, that the dissection concept is developed with considerable effort for nonlinear DAEs in [15].

A.4. Admissible matrix function sequences and related subspaces
In this part we apply several routine notations and tools used in the projector based analysis of DAEs. We refer to the appendix for a short roundup and to [17,23] for more details.
Given are at least continuous matrix functions E, F : I → R m×m , E has a C 1 -nullspace and constant rank r. We use a proper factorization E = AD where A : I → R m×k , D : I → R k×m , and B = −(F + AD ′ ). R : I :→ R k×k denotes the continuously differentiable projectorvalued function such that im D = im R and ker A = ker R.
Let Q 0 : I → R m×m denote any continuously differentiable projector-valued function such that im Q 0 = ker D = ker E, for instance, Q 0 = I − D + D with the pointwise Moore-Penrose inverse D + . Set P 0 = I − Q 0 and let D − denote the pointwise generalized inverse of D determined by Set G 0 = AD, B 0 = B, Π 0 = P 0 . For a given level κ ∈ N, the sequence G 0 , . . . , G κ is called an admissible matrix function sequence associated with the pair {E, F} and triple {A, D, B}, respectively, e.g., [17,Definition 2.6] if it is built by the rule The admissible matrix functions G i are continuous. The construction is supported by two constant-rank conditions at each level. It results that 0 < r T 0 ≤ r T 1 ≤ · · · ≤ r T i ≤ . . . , 0 ≤ u T 1 ≤ u T 2 ≤ · · · ≤ u T i ≤ . . . By construction, the inclusions are valid pointwise. There are several special projector functions incorporated in an admissible matrix function sequence, among them admissible projectors Q i onto ker G i and Π i = Π i−1 (I − Q i ), Π 0 = (I − Q 0 ), yielding the further inclusions ker Π 0 ⊆ ker Π 1 ⊆ . . . ⊆ ker Π r = ker Π r+1 .
Each of the time-varying subspaces in (53) and (54) has constant dimension, which is ensured by the respective rank conditions. The subspaces involved in (53) and (54) are proved to be invariant with respect to special possible choices within the construction procedure and also with respect to the factorization of E = AD. Definition A.1. The DAE given by the coefficient function pair {E, F} or a related tripel {A, D, B} is called regular if there are an index µ and an admissible matrix function sequence G 0 , G 1 , . . . , G µ such that r T µ = m. The tractability index of the DAE is defined to be the smallest index µ with r T µ = m, and the integers 0 < r T 0 ≤ r T 1 ≤ · · · ≤ r T µ−1 < r T µ = m are called characteristic values of the regular DAE.