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Reduction and Normal Forms of Matrix Pencils

Matrix pencils, or pairs of matrices, may be used in a variety of applications. In particular, a pair of matrices (E,A) may be interpreted as the differential equation E x' + A x = 0. Such an equation is invariant by changes of variables, or linear combination of the equations. This change of variables or equations is associated to a group action. The invariants corresponding to this group action are well known, namely the Kronecker indices and divisors. Similarly, for another group action corresponding to the weak equivalence, a complete set of invariants is also known, among others the strangeness. We show how to define those invariants in a directly invariant fashion, i.e. without using a basis or an extra Euclidean structure. To this end, we will define a reduction process which produces a new system out of the original one. The various invariants may then be defined from operators related to the repeated application of the reduction process. We then show the relation between the invariants and the reduced subspace dimensions, and the relation with the regular pencil condition. This is all done using invariant tools only. Making special choices of basis then allows to construct the Kronecker canonical form. In a related manner, we construct the strangeness canonical form associated to weak equivalence.

1. Introduction 1.1. Equivalence. The primary study of this paper is that of pairs of matrices, also called matrix pencils. In other words, we study pairs of operators (E, A) both acting from a finite dimensional vector spaces M to a finite dimensional vector space V .
A typical example we have in mind is the linear differential equation Such a model is clearly invariant by changes of variable, or by changing the order of the equations. More precisely, it is invariant by simultaneous equivalence transformation of the operators E and A. The corresponding equivalence relation is the following: two pairs of operators (E 1 , A 1 ) and (E 2 , A 2 ) will be considered equivalent if there exists invertible operators P and Q, operating on M and V respectively, such that E 2 = PE 1 Q, This equivalence relation is associated to a group, which is simply GL(M )×GL(V ). This is called strong equivalence in [1]. We are interested in properties which are invariant with respect to that group action on the matrix pencil. In other words, we are interested in quantities that label the orbit of the group action.
In fact, a complete set of invariants and a canonical form have been known since the works of [16] and [6]. Modern versions of those proofs may be found in [1,§ XII.4] and in [2, § A.7]. The primary tool for obtaining those invariants is the Jordan canonical form. For that reason, those proofs are impossible to extend to nearby cases, for example to the infinite dimensional case, or to the parameter dependent case, not to mention the numerical difficulties associated with the computation of the Jordan canonical form.
As a result, alternative proof techniques were developed, most notably in [13] and [17]. Those authors observed indeed that using a Jordan canonical form is not suitable to compute the invariants other than the Jordan invariants, i.e., the Kronecker indices and the "infinite elementary divisors" [13]. The idea is to transform the pair of matrices into a form which exhibits all the invariants but is not a canonical form. Those forms are known under the names of generalized Schur-staircase form, or GUPTRI (Generalized Upper Triangular Form). We refer to [4, §4.1] and [8] for more references on those algorithms.
Our approach is similar, although with a shift of focus towards the underlying algebraic structures as opposed to the algorithmic aspects. In particular, we attempt to define the invariants from the dimensions of subspaces which are themselves invariants with respect to the equivalence relation at hand. The advantage of our approach is that a great deal of results are automatically independent of the choice of a basis, or any other structure (like a Euclidean structure).

Invariants.
A matrix pencil, when considered as a differential equation (1), may be decomposed in an intrinsic ordinary differential equation, and an extra structure. We will call the invariants of the underlying ordinary differential equation the dynamical invariants, and we will call the remaining invariants the nondynamical invariants. In the parlance of the Kronecker decomposition theorem as presented in [1], the dynamical invariants would be the finite elementary divisors (essentially a Jordan form), whereas the non-dynamical invariants would be the infinite elementary divisors along with the row and column minimal indices.
The dynamical invariants, i.e., the invariants of the intrinsic differential equations boil down to the Jordan invariant associated to similarity transformations, and are therefore of less interest to us. We will thus mostly focus on the non-dynamical invariants, which appear only when E is not invertible. Those invariants are wellknown in control theory, and in the study of differential algebraic equations. In control theory, such invariants are the controllability and observability indices ([5, § 6.3]), for differential algebraic equations (DAE), in the case of a regular pencil (see subsection 3.8), the most used non-dynamical invariant is the index ([3, VII.1]). Our goal is to define the non-dynamical invariants in a invariant manner, without any other structure than the linear algebraic structure.
1.3. Reduction. The crucial tool to the study of the invariants of a pencil is the concept of reduction, which we define precisely in Section 2.
This concept was gradually developed, under various names, or no name at all, first in [18] for the study of regular pencils, then in [17, §4] and [13] to prove the Kronecker decomposition theorem. It is also related to the geometric reduction of nonlinear implicit differential equations as described in [10] or [9]. In the linear case, those coincide with the observation reduction, as shown in [14]. It is also equivalent to the algorithm of prolongation of ordinary differential equation in the formal theory of differential equations, as shown in [12].
The reduction procedure is an operation that, out of a pair of operators (E, A), creates a new, smaller one (E , A ). "Smaller" is in the sense that the reduced operators E and A are restrictions of E and A on subspaces of M and V , defined by V := EM and M := A −1 V .
This process of reduction is iterated, producing systems (E (k) , A (k) ) and subspaces M (k) and V (k) . This process ultimately stops, and we will call the number of steps before it stops the index.
When the process stops, the system which is produced, denoted by (E (∞) , A (∞) ), is such that E (∞) is surjective. After running the reduction algorithm once more on the dual of that reduced system, i.e., on (E (∞) * , A (∞) * ), one obtains an isolated system ( At each step of the reduction, some information from the original system is lost. That information is encoded by integers called "defects". Those defects are of three kinds: α, β + and β − . The defect α 1 is defined as the dimension of the kernel of E, regarded as a quotient operator from M/M to V /V . The defect β + 1 is defined as the dimension of the cokernel of A, regarded as a quotient operator from M/M to V /V . The iterated reduction then generates the sequences of defects α k and β + k . The defects β − k are defined as the β + defects of the system (E (∞) * , A (∞) * ). Using those subspaces, defined in an invariant manner, we are able to show the following facts: • the operator E is invertible if and only if all the defects vanish • the pair (E, A) is a regular pencil if and only if the β + and β − defects vanish • we show that the invariants defined in [7], like the strangeness, may also be defined directly in an invariant manner, i.e., without using any extra structure or basis We also show that the defects and the system (E (∞) * (∞) , A (∞) * (∞) ) completely characterize the equivalence class corresponding to the equivalence relation (2).
• the defects are related to the Kronecker indices • the invariants defined in [7] may be used to construct a corresponding canonical form for weak equivalence: this connects the approaches of [17] and [7] • using the relation with the Kronecker decomposition theorem, we show that the defects of the dual system (E * , A * ) are related to those of (E, A) by switching the β + and β − defects.

1.4.
Outline. The layout of the paper is as follows.
In the first part, Section 2 and Section 3, we show how to derive the nondynamical invariants. In Section 2 we define the reduction procedure. In Section 3 we define the defects of a system, and study their properties. In particular, we give an original proof of the relation between the property of a pencil to be regular, and the presence of some of the defects.
In the second part, we show that the invariants obtain in the first part, namely the defects, supplemented by a Jordan structure, are the only invariants of the pair of matrices with respect to equivalence. Most of the results in this part are already in [17] and [13]. In Section 4 we prove the basic lemmas needed to construct canonical forms. In Section 5, we show how to use those tools to construct a canonical form with respect to weak equivalence In Section 6 and Section 7 we show that the defects determine a complete canonical form. In Section 8 we study the relation with the existing Kronecker canonical form.

System Reduction
Definition 2.1. We will call a pair of linear operators (E, A) a linear system, or simply a system, if E and A have the same domain and codomain, both of finite dimension.
Given a system (E, A), we will denote the common domain of E and A by M (E,A) and the common codomain of E and A by V (E,A) , so a system (E, A) may be represented as

2.2.
Reduced spaces. The idea behind the reduction of a linear system (E, A) is to "disentangle" the spaces associated with the operators E and A. The strategy pursued is to try and make the operator E surjective, by successive reduction steps. In order to achieve this, we have to describe the lack of surjectivity of E, first independently of A, which leads to the definition of the subspace The next step is now to describe the lack of surjectivity of E, with respect to A, which we measure using the subspace Remark 2.2. Those definitions make sense when considering the differential equation Ex + Ax = 0.
Notice that any suitable initial condition for this equation must be in M . If the initial condition is not in M , there cannot be any solution stemming from that initial condition.
Let us put those definitions together: Remark 2.4. We will often drop the dependency on the system (E, A), and simply write M , M , V and V when the context is clear enough.
Remark 2.5. As explained in [14, §5.1], the reduction of Definition 2.3 corresponds to the non-linear reduction of general systems of differential equations with constraints. The study of differential equations is also the point of departure in [17].
Remark 2.6. One of the first occurrence of the definition of that subspace M seems to be in [18,Lemma 2.1]. It is used to study systems which are regular pencils (see Definition 3.19). Another explicit definition is to be found in [11, §7], although with a different purpose than ours, namely the study of linear, time-varying differential algebraic equations of index one.
2.3. System Reduction. The subspaces M (k) and V (k) defined in Definition 2.3 allow for defining a new system. This procedure will be called "reduction".
Proposition 2.7. Given a system (E, A), the operators E and A are uniquely defined by the following commuting diagram.
The vertical arrows are canonical injection from a subspace into the ambient space.
The operators E and A build up a new system (E, A) which we call the reduced system, and is defined by Proof. The proof rests on the observation that Remark 2.8. Consider the category which objects are vector spaces and arrows are systems as defined in Definition 2.1. The reduction operation, denoted by a prime, is an endofunctor in this category, i.e., a functor from that category to itself.
As we mentioned in the beginning of subsection 2.2, our goal is to obtain a reduced system such that E is surjective. It is only part of a general strategy to obtain a reduced system where E is invertible. It is therefore important that the reduction algorithm does not alter the injectivity of E. We observe that this is indeed the case. Proposition 2.9. If, in a system (E, A), E is injective, then E is also injective.
Proof. It is a consequence of the observation that

Iterated Reduction.
We may iterate the reduction process described in subsection 2.3 on the new system (E, A) . This leads to a sequence of systems {(E, A) (k) } k∈N which is defined recursively as follows.
Definition 2.11. The iterated of the reduction of a system (E, A) are defined recursively by We will make use of the straightforward notation, for k ∈ N. (3) . The reduced operators E (k) and A (k) are essentially restrictions of the original operators E and A, so we may rewrite the definition of the iterated reduced subspaces M (k) and V (k) . Proposition 2.12. For a system (E, A) the following assertions hold for any integer Proof. The proof is a simple verification by induction on k.
2.5. Totally Reduced Systems. As we shall notice in subsection 2.7, the repeated operation of reduction transforms a system into one which cannot be reduced anymore, or rather, for which the reduction does not create a new system. We call such systems "totally reduced": Definition 2.13. We will say that a system (E, A) is totally reduced if (E, A) = (E, A).
A practical characterisation of a totally reduced system is that V = V . The verification is straightforward.
2.6. Almost Reduced System. Definition 2.15. We will say that a system (E, A) is almost reduced if The chosen vocabulary is supported by the following facts: • a totally reduced system is also almost reduced, which follows from Definition 2.3. • a system which is almost reduced will be totally reduced at the next step of the reduction, since by Proposition 2.12: Remark 2.16. In the situation of a reduced system which is almost but not totally reduced, the following subspace sequences would be produced.
A concrete example where this happens is when A = 0 and E is not surjective. It is clear that M = M but V V . The corresponding system is thus almost reduced but not totally reduced.

2.7.
Index. The reduction procedure produces decreasing sequences of subspaces. When both sequences stall, the system is totally reduced. The number of reduction steps needed to transform a system into a totally reduced one is called the index of the system (E, A): Definition 2.17. The smallest integer n ∈ N for which the system (E (n) , A (n) ) is totally reduced is called the index of the system (E, A).
We will use the following notation for the index of the system (E, A): Remark 2.18. The index is always a finite integer 1 , and the reduced system (E , A ) has an index dropped by one, i.e, Those observations will be used repeatedly to prove statements by induction on the index (e.g., in Proposition 3.20, Theorem 6.1 and Theorem 8.2).
Remark 2.19. Using Proposition 2.14 we observe that Remark 2.20. The index defined in Definition 2.17 is closely related to the geometric index defined in [10], [9] or [14, §5.1]. In fact, the geometric index would be the first integer n such that the system (E, A) (n) is almost reduced (Definition 2.15). As we shall see in Corollary 3.17 and Proposition 3.20, this minor difference is only relevant for singular pencils.
Remark 2.22. We could simply have defined, say E (∞) by the limit of the sequence of operators E (k) (because this sequence eventually stalls), which explains the notation "∞".
We pointed out in subsection 2.2 that the idea behind the reduction procedure was to lead to a system where E is surjective. The reduction algorithm indeed achieves this goal: Proposition 2.23. The totally reduced operator E (∞) is surjective. Proof. The system (E (∞) , A (∞) ) is totally reduced so we may use Proposition 2.14 to conclude that At each step of the reduction some information is lost, by passing from the original system to the reduced one. We capture that information loss by two quotient operators defined on the quotient space M/M .

Constraint and Observation Defects. Since [A] is injective and [E] is sur
Now we take advantage of the reduction procedure and define those defects recursively: Similarly, the observation defects β + k (E, A) are defined for any integer k ≥ 1 by

Control Defects.
There is another important kind of defect that will be needed. In is obtained by considering the dual of the totally reduced system obtained after repeated reductions. That totally reduced system ( such that E is injective? It turns out that such a system has no constraint defects. Proposition 3.4. Given a system (E, A), if E is injective, then the system has no constraint defects, i.e., for all integer k ≥ 1, α k (E, A) = 0.
Proof. The proof proceeds by induction on the index. (2) For a positive index, using Proposition 2.9 we may apply the induction hypothesis and deduce that α k (E, Let us introduce the notion of a dual system. Notation 3.5. Given a system (E, A) we define the dual system (E, A) * by the pair of adjoint operators (E * , A * ), i.e., Proposition 3.6. The dual (E (∞) , A (∞) ) * of a totally reduced system has no constraint defects, i.e., Proof. According to Proposition 2.23, the operator E (∞) is surjective, so E (∞) * is injective, and we conclude using Proposition 3.4.
This suggests that another set of defects is given by the observation defects of the dual of the totally reduced system (E (∞) , A (∞) ).
3.4. Intrinsic Dynamical System. The reduction procedure may thus be used once to obtain a totally reduced system, and may then be applied again to the dual of that totally reduced system. Starting with a system (E, A), we may completely reduce it to obtain the system (E (∞) , A (∞) ). The operator E (∞) * is injective. The adjoint system (E (∞) * , A (∞) * ) may be in turn completely reduced to obtain the system (E (∞) * (∞) , A (∞) * (∞) ). Using Proposition 2.23 and Proposition 2.9, we obtain the following result.
Since the operator E (∞) * (∞) is invertible, its domain and co-domain have the same dimension. This dimension is the dimension of the intrinsic dynamics of the system. Definition 3.9. The dynamical dimension δ of the system (E, A) is defined by the integer Remark 3.10. For a differential equation defined by the system (E, A), the system corresponds to the underlying differential equation. In particular, the dynamical dimension δ determines the degrees of freedom for the choice of the initial condition.
3.5. Dimensions of the Subspaces. In order to study the relations existing between the defects and the various subspaces M (k) and V (k) , we define the following spaces, which measure the difference of dimension between each successive reduction: Definition 3.11. Recalling Definition 2.11, for any integer k ≥ 1 we define the spaces By definition of the defects in Definition 3.2 and using Proposition 3.1, one obtains the relations between the dimensions of the spaces defined in Definition 3.11 and the defects. For any integer k ≥ 1 this implies the inequalities Remark 3.12. This is the same sequence of inequalities as in [17, 5.2].
In particular, the dimensions of the spaces ∆M (k) and ∆V (k) may be expressed using the constraint and observation defects.
Proof. Those identities follow from an induction based on (6) and the observation that the integers dim ∆V (k) and dim ∆M (k) are zero when k is bigger than the index of the system. Remark 3.14. As we shall see in Theorem 5.1, the quantity defined in [7] as the "strangeness" s turns out to be the integer Roughly speaking it expresses the number of constraints that, when differentiated, will help to reduce the system. We may thus give the precise relation of the strangeness to the defects using Lemma 3.13, namely The dimensions of the spaces M and V may also be expressed from the defects and the dynamical dimension δ (see Definition 3.9).
Proposition 3.15. The dimensions of M , V , the defects α, β + and β − and the dynamical dimension δ are related by the formulae Using Lemma 3.13 we obtain Using Lemma 3.13 we thus obtain dim ∆V (k) = 0 ⇐⇒ α j + β + j = 0 ∀j ≥ k + 1, which proves the claim.
In the case of a system without observation defects we obtain readily: 3.7. Defects and Invertibility. The choice of the name "defect" may seem overly negative, but those integers really measure how far this system is from a system where E is invertible. This is the essence of the following proposition.
Proposition 3.18. For a given system (E, A) the following statements are equivalent.
(ii) The operator E is invertible.
Proof. E is surjective if and only if ∆V = 0. By Lemma 3.13, that is equivalent to α = β + = 0. Since E is invertible if and only if both E and E * are surjective, we obtain the result using Definition 3.7.

Regular Pencils.
A pencil is a polynomial on a ring of matrices. Since we are interested in pairs of matrices, our attention is restricted to first order polynomials, and to the property of such a polynomial to be regular.
There is a remarkable relation between the property of being regular and the defects: We need first a lemma to understand how the pencil regularity property may be lost during the reduction. (1) Consider, for any λ ∈ C, the operator S λ defined by S λ can be decomposed into S λ and [S λ ] according to the following commuting diagram: Since both rows are exact sequence, out of the three operators S λ , S λ and [S λ ], if two of them are invertible then the third one is. One easy way to prove this fact 2 is by choosing bases in M and V which are compatible with the subspaces M and V . The operator S λ is then represented by a block triangular matrix where the diagonal blocks are the matrices of S λ and [S λ ]. Now it is easy to check that if two of those three matrices are invertible, the third one is.
is a regular pencil, and the claim is proved.

4.1.
Motivation: coupling spaces. In Section 2 we showed how to define invariant subspaces for the system (E, A). "Invariant" means here that those subspaces are not arbitrarily chosen, they depend in a unique way from the system at hand. In order to obtain a simple matrix representation of that system, we will need to choose supplementary spaces to the invariant subspaces M and V . In this section, we focus on such supplementary spaces for one reduction step only, and establish some results which will be needed in Section 6.
We first look at the case of supplementary subspaces to the subspace M , i.e., subspaces N ⊂ M such that M = M ⊕ N . The strategy is to try and choose N in the same direction as the part of ker E that remains out of M . First we define what this space is by decomposing the kernel of E in the part that is included in M and some supplementary space. This is achieved by choosing any supplementary space K such that Then since, by construction, K ∩ M = 0 one may complete M by choosing a supplementary space C such that We now define N as N := C ⊕ K . The choice of C will prove to be essential to obtain a complete decomposition of the system (E, A). The tool to choose C appropriately will be Lemma 4.2.
But notice now that no matter how we choose C , the space K roughly speaking corresponds to the variables that are decoupled from the rest of the system. They are sometimes called the algebraic constraints. The variable y is decoupled from the rest of the system.

4.2.
Coupling Lemma for E. We will assume that some coupling space W has already been chosen in the reduced system, and that will serve as a starting point for the choice of the coupling space at the present stage. More precisely, we assume that the reduced space V is already decomposed as That decomposition allows to construct the coupling spaces in an optimal manner, in one subspace C coupled with W , and complement with vectors in the null-space of E. We state the result in a lemma, formulated outside the context of linear systems. and such that the sequence is exact. The exactness means here that ker E ∩ (K ⊕ C ) = K and E(K ⊕ C ) = W .
Moreover, for any choice of basis in W one may choose a basis of C such that its image by E is the basis in W (see Figure 1).
We conclude that C ∩ M ⊂ ker E.
Those subspaces are such that the following sequences are exact: and such that AM ∩ Z = 0. Moreover, one may choose basis in the subspaces C , K , D and Z such that the basis of D is the image by A of the basis of N , and the basis on W is the image by E of the basis of C .

Proof.
(1) By the assumption on W , we have The subspace W is moreover equipped with a basis by the induction hypothesis. Note that, given a basis in W we can choose a basis on C such that E sends that basis on that of W . Let us now define the subspace N ⊂ M by We choose an arbitrary basis of the space K , and this provides us with a basis for the space N .

Strangeness
In order to illustrate the power of reduction, and to show an application of Lemma 4.4, we show an intermediate result. Instead of looking at the equivalence classes for the equivalence of matrices, that is, pairs of invertible operators acting on (E, A) as (PEQ, PAQ), we look at the weak equivalence.
Weak equivalence is determined by another group, which elements consist of two invertible operators P and Q and an arbitrary operator R, acting on a system (E, A) as (9) (P, Q, R) · (E, A) := (PEQ, P(ER + AQ)).

5.1.
Weak equivalence group. The group operation corresponding to weak equivalence is given by where P 1 , P 2 are automorphisms of V , Q 1 , Q 2 are automorphisms on M , and R 1 , R 2 are arbitrary endomorphisms on M . The identity is then (I, I, 0), and the inverse of an element (P, Q, R) is given by Clearly, the elements of the form (P, Q, 0) form a subgroup corresponding to the equivalence relation. Another subgroup is given by elements of the form (I, I, R).
For the study of the orbits of the weak equivalence group, the identity shows that we may restrict our attention to one subgroup at a time.

Orbit Invariants.
The orbits of the weak equivalence group action (9) were studied in [7], in which the authors exhibited a complete set of invariants. We give an alternative proof here, thereby shedding some light on the notion of strangeness.
Theorem 5.1. A complete set of invariants for the group action (9) is given by The integer s is called "strangeness" in [7].

Proof.
(1) First we have to check that the three integers are indeed invariants of the group action. Clearly, they are invariants by transformations of the form (P, Q, 0), which are merely equivalent transformation.
Let us examine the case of a transformation (E, A) = (I, I, R) · (E, A) = (E, ER + A). (10), this shows that the spaces V , V and M are invariants of all of the weak equivalence group transformations.

We have
As a result, the spaces ∆V and the operator [E] are also invariants, so the integers d, a and s are invariants.
(2) Now we show that the integers d, a and s are the only invariants. In order to show that, we show that a system (E, A) is weakly equivalent to a canonical form that depends only on those three integers. In order to achieve this, we decompose M and V using Lemma 4.4. 6. Direct Decomposition 6.1. Decomposition Theorem. In Section 2 we showed how to define invariant subspaces M (k) and V (k) for the system (E, A). In order to obtain a complete decomposition of the spaces M and V , it is necessary to construct subspaces that bridge the gap between each invariant subspaces M (k) and V (k) . More precisely we construct spaces N (k) and W (k) such that In a sense, the subspaces N (k) and W (k) correspond to the spaces ∆M (k) and ∆V (k) respectively, defined in Definition 3.11. The construction of those supplementary spaces proceeds backwards, in the direction opposite to the reduction. One must first totally reduce the system (E, A). Assume that the index is n. One then chooses an arbitrary complementary space W (n) such that V (n−1) = V (n) ⊕ W (n) , and equip that space with an arbitrary basis. The rest is a repeated application of Lemma 4.4.
We now see how to choose those supplementary spaces N (k) and W (k) so that the operators E and A are simultaneously decomposed in an advantageous way.
Recall the definitions of the subspaces M (k) and V (k) in (3).
For any integer k ∈ N there exists subspaces N (k+1) ⊂ M and W (k+1) ⊂ V such that and for any integer k ≥ 1 there exists subspaces such that Those subspaces are such that for any integer k ≥ 1, the following sequences are exact (see Figure 3): Moreover, one may choose basis in the spaces C (k) , K (k) , D (k) and Z (k) such that the basis of D (k) is the image by A of the basis of N (k) , and the basis on W (k+1) is the image by E of the basis of C (k) . and V (∞) the matrix is block diagonal (see Corollary 6.4), we may choose those bases in such a way that E is represented as the identity matrix on that block. Since we now that E (∞) is surjective (Proposition 2.23), that identity block stretches to fill V (∞) .
Remark 6.2. For the reader averse to the language of exact sequences, the fact that the sequences of Theorem 6.1 are exact means in that case that Remark 6.3. In the same spirit as Remark 4.5, we notice the relation between the dimensions of the various subspaces introduced in Theorem 6.1, and the dimensions of the spaces defined in Definition 3.11, and to the defects (Definition 3.2). For any integer k ≥ 1, the relations are given by Proof of Theorem 6.1. We proceed by induction on the index (see Figure 4). If the index is zero, all the spaces M (k) and V (k) are zero, and there is nothing to prove. Assume now that the statement holds for systems of index n − 1. Given a system (E, A) of index n, the reduced system (E , A ) has index n − 1, so we may apply the induction hypothesis on that reduced system.  Starting with W , one constructs the space C such that EC = W , and a subspace K such that K ⊂ ker E using Lemma 4.2, and defines N := C ⊕ K . One then constructs Z such that V = V ⊕ AN ⊕ Z . This in turn defines W := AN ⊕ Z .
The reduced system (E, A) consists of operators operating from M to V , so by the induction hypothesis we obtain a decomposition of the spaces M and V into subspaces W (k) , N (k) as described in the statement of the theorem.
We have to shift the indices of all the spaces produced for the final statement to hold. For example, we define for any integer k ≥ 2 so we may write the decomposition of V as The reduced operators E and A being restrictions of E and A, the statements obtained from the induction hypothesis apply to the operators E and A.
Applying Lemma 4.4 yields the desired result.

Decomposition in invariant subspaces.
A crucial consequence of Theorem 6.1 is that it provides us with decompositions of M and V such that E and A may be restricted on those subspaces: and we have Proof. The fact that EM ⊂ V and AM ⊂ V follows from 7. Dual Decomposition 7.1. Dual space decomposition. Assume that a finite dimensional vector space M is decomposed in a direct sum of subspaces, i.e., This decomposition induces the dual space decomposition Although it is not reflected by the notation, it is clear that (M k ) * actually depends not only on M k but on all the other spaces of the decomposition. It is a generalization of the notion of dual basis.
Assume further that M is equipped with a basis. We say that this basis is compatible with the decomposition if each subspace is the span of a subset of the basis.
If M is equipped with a basis B compatible with a subspace decomposition, then the dual basis is compatible with the dual space decomposition.
Indeed, consider a subspace M k of the decomposition. Since the basis is compatible with the decomposition, that subspace is spanned by a subset of the basis B, say S M k ⊂ B, i.e., The dual decomposition is such that the associated subspace (M k ) * is the span of the dual basis with the same subset S M k , i.e., (M k ) * = span e * : e ∈ S M k .
Here the covector e * is the element of the dual basis of B corresponding to e, i.e., such that Lemma 7.1. Assume that A and B are subspaces of M that are part of a subspace decomposition of M , and that C is a subspace of V that is part of a subspace decomposition of V . Assume further that M and V are equipped with bases compatible with their decompositions. Take an operator S operating from M to V . The following two statements are equivalent: (i) The following sequence is exact: Moreover, the operator S sends the basis of B on the basis of C. (ii) The following "dual" sequence is exact: Moreover, the operator S * sends the basis of C * on the basis of B * .
Proof By applying Theorem 6.1 on the dual system (E * , A * ) one obtains a decomposition of M and V as Moreover, all those subspaces are equipped with a suitable basis. By choosing a basis for the spaces M For any integer k ∈ N we have Those subspaces are such that the following sequences are exact (see Figure 3): Proof. It is a direct application of Lemma 7.1.
Remark 7.3. The exact sequences of Theorem 6.1 are the same as those of Theorem 7.2 but with flipped arrows. It just reflects how the block structure of a matrix is related to the block structure of the transposed matrix. 7.3. Second sweep of the decomposition. Remember that the constraint defects of the adjoint of the totally reduced system (E (∞) * , A (∞) * ) are zero (Proposition 3.6). Along with Remark 6.3, we conclude that the corresponding subspaces K (k) produced in Theorem 6.1 are zero. Now, using Corollary 6.4, we are in a position to use the decomposition of Theorem 6.1 for the dual system (E (∞) * , A (∞) * ) and obtain a decomposition of M (∞) * and V (∞) * . Theorem 7.4. In addition to the decomposition given by Theorem 6.1, the spaces M (∞) and V (∞) may now be decomposed as (see Figure 5): Remark 7.5. The various defects defined in Definition 3.2 and Definition 3.7 may now be pictured clearly using the Theorem 7.4; see Figure 6.

Kronecker Indices
8.1. Basis Arrangement. In this section we prove a result on the basis obtained in Theorem 6.1, which will be useful to determine the relation with the Kronecker decomposition theorem. Notice that this block is completely separated from the rest, so one may now reduce the A to Jordan blocks by a similarity transformation.
Definition 8.1. For k ∈ N, k ≥ 1, we define a N k -sequence to be a sequence m j ∈ M 1 ≤ j ≤ k of k independent vectors in M , and a sequence v j ∈ V 1 ≤ j ≤ k of k independent vectors in V such that Am j = v j for 1 ≤ j ≤ k, Em j = v j−1 for 2 ≤ j ≤ k, Em 1 = 0 and v k ∈ Im E, which is summarized in the following diagram.
Similarly, for k ∈ N, k ≥ 1, we define a L k -sequence to be a sequence m j ∈ M 1 ≤ j ≤ k − 1 of k − 1 independent vectors and a sequence v j ∈ V 1 ≤ j ≤ k of k independent vectors which fulfill the conditions summarized in the following diagram.
Theorem 8.2. Theorem 6.1 produces bases such that there are α k N k -sequences, and β + k L k -sequences. Moreover, the end vectors v constitute a basis of W . Proof.
(1) We proceed by induction on the index. Assume that the result holds for the reduced system (E , A ).
(2) The basis of C is precisely such that Em k+1 = v k . Besides, the basis of AC is chosen such that v n+1 = Am k+1 , so each L k and N k sequence is extended with two elements, meaning that they build now L k+1 and N k+1 sequences. (3) Each element m of the basis of K produces a new N 1 sequence (m, Av), since Em = 0 and Am = v ∈ Im E: The dimension of K being α 1 , we produce α 1 such sequences. (4) Each element v of the basis Z qualifies as a L 0 sequence, since v ∈ Im E and v ∈ Im A, so Since the dimension of Z is β + 1 , we produce β + 1 such sequences.
The nilpotent blocks N E k E part of the nilpotent blocks and N A k A part of the nilpotent blocks defined by Theorem 8.5. A decomposition with defects α, β + , β − , produces a Kronecker decomposition which for all integer k ≥ 1 contains • α k block of type N k , • β + k blocks of type L k , • β − k blocks of type L T k . Proof. Recall the definition of L k and N k sequences in Definition 8.1. By regrouping the elements of an L k sequence, one obtains a representation of E and A as a L kblock, and similarly, by regrouping the elements of a N k -sequence, one obtains a N k block. Applying Theorem 8.2, and regrouping the basis elements stemming from the sequences L k and N k we obtain α k N k -blocks and β + k L k -blocks, for k ≥ 1. Now the basis on the sub-block M (∞) , V (∞) are obtained by transposing the decomposition given by Theorem 6.1. Using the previous step and Proposition 3.6, we obtain β − k transpose of L k -blocks, for k ≥ 1. Remark 8.6. It is remarkable that the decomposition obtained in Theorem 7.4 produces basis vectors which are the same as for a Kronecker decomposition, only ordered differently. The necessary permutations may be visualised on Figure 6. 8.3. Conjugate Decomposition. We may now show the relation between the defects α, β + and β − of a system (E, A) and the defects of the adjoint system (E * , A * ). It turns out that the constraint defects are the same and that the observation defects β + and the control defects β − are just switched. This fact would have been very difficult to prove from the results of Section 3 alone, so we need the full power of Theorem 7.4 and of its consequence, Theorem 8.5.
Theorem 8.7. The conjugate decomposition switches the defects β + and β − , i.e., it produces the defects α(E * , A * ) = α(E, A), Proof. It is a consequence of Theorem 8.5, for when putting the system in Kronecker form Definition 8.4 and transposing, the system is still in Kronecker form, but the bidiagonal blocks L and (L) T are switched.
8.4. Weierstraß decomposition. In the case of regular pencils (see subsection 3.8), the Kronecker decomposition is called the Weierstraß decomposition ( [16,1]) and is such that E and A take the matrix representation where C may be in Jordan normal form and N is a block diagonal matrix of blocks of type N E k . The matrix block NMatrix of nilpotent blocks is a diagonal block matrix N = diag N E k1 (0), N E k2 (0), . . . , N E km (0) where the blocks (N E k1 (0) are the nilpotent blocks defined in Definition 8.3. Corollary 8.8. The Weierstraß decomposition is such that for any integer k ≥ 1 it contains α k blocks N E k . Proof. It is just a special case of Theorem 8.5 using Proposition 3.20.

Conclusions
We have defined the notion of defects and have related them to existing concepts, such as the regular pencil condition, the dimension of the reduced subspaces, or the notion of strangeness. We also showed how the defects define a normal form, and how that normal form relates to the existing one of Kronecker.
Note that some results, as Theorem 8.7, would be difficult to prove without using the canonical form. Nevertheless, we tried to wring the most out of the invariant objects defined in Section 3.
The advantage of such an approach is that it is extensible to nearby cases such as the parameter dependent case, or the infinite dimensional case (see [15]).