Linear relations and the Kronecker canonical form

Linear relations and the Kronecker canonical form Thomas Berger Carsten Trunk Henrik Winkler August 5, 215 Abstract We show that the Kronecker canonical form (which is a canonical decomposition for pairs of matrices) is the representation of a linear relation in a finite dimensional space This provides a new geometric view upon the Kronecker canonical form Each of the four entries of the Kronecker canonical form has a natural meaning for the linear relation which it represents These four entries represent the Jordan chains at finite eigenvalues, the Jordan chains at infinity, the so-called singular chains and the multi-shift part Or, to state it more concise: For linear relations the Kronecker canonical form is the analogue of the Jordan canonical form for matrices Keywords: Linear relations, Jordan chains, Kronecker canonical form, Wong sequences AMS subject classifications: Primary 15A21, 47A6; Secondary 15A22, 47A15 1 Introduction Solutions of linear ordinary differential equations of the form ẋ(t) = Ax(t) can be completely characterized by the eigenvalues and generalized eigenvectors of the matrix A, ie, by the Jordan canonical form of A If E is an invertible matrix, the same applies to the equation and the solutions are encoded in the Jordan canonical form of Eẋ(t) = Ax(t), (11) E 1 A (12) The situation is more challenging when E is not invertible Then (11) may contain purely algebraic equations; for instance if E has a zero row, then the corresponding equation does not contain any derivatives Thus (11) is called a differential-algebraic equation (DAE), see eg [1, 18, 2 A characterization of the solutions of (11) (see eg [6) is done via the Kronecker canonical form (KCF ) The KCF is a canonical form for a matrix pair (E, A) (often considered in the form of a matrix pencil se A) and, hence, a generalization of the Jordan canonical form It has its origin in [17, see also [15 But even in the case of a non-invertible matrix E, the expression (12) can be given a meaning For this we use the theory of linear relations (or, what is the same, of multi-valued mappings), see [1, 26 for instance Each matrix (or linear mapping) is considered via its graph as a subspace in C n C n thomasberger@uni-hamburgde, Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 2146 Hamburg, ermany carstentrunk@tu-ilmenaude, henrikwinkler@tu-ilmenaude, Institut für Mathematik, TU Ilmenau, Weimarer Straße 25, 98693 Ilmenau, ermany 1

Addition and multiplication of two subspaces are defined in analogy to the addition and multiplication of two linear mappings, for details see Section 2 In the sense of linear relations, the inverse E 1 of a non-invertible matrix E is given as the subspace of all tuples ( Ex x ) in Cn C n Then expression (12) has a natural meaning, E 1 A = { (x, y) C n C n Ax = Ey }, which was already studied in [4, 5 An eigenvector at λ C of E 1 A is a tuple of the form (x, λx) E 1 A, x, and thus satisfies Ax = λex It follows that the (point) spectrum of E 1 A and the (point) spectrum of the matrix pencil se A coincide This shows a deep connection between the matrix pair (E, A) and the linear relation E 1 A It is the aim of the present paper to explore this connection While the connection between the (point) spectra (and the Jordan chains) of E 1 A and the matrix pencil se A is quite obvious and in a certain sense due to the right definition of Jordan chains of E 1 A, another aspect is more stunning and the main objective of this paper: The connection of the KCF of a matrix pair and the linear relation A represented with the help of these matrices For this we show in Section 3 that an arbitrary linear relation A in C n C n can be represented with matrices A, E, F, in the following way A = F 1 = E 1 A (13) We restrict ourselves to the representation A = F 1 and show that the KCF of the matrix pair (F, ) is indeed a canonical form for the linear relation A We show that each of the four entries of the KCF has a natural meaning for the linear relation A These are the Jordan chains at finite eigenvalues, the Jordan chains at infinity, the singular chains and the multishifts This is the main result of the present paper: The Kronecker canonical form of a matrix pair (F, ) is the canonical form for the linear relation A = F 1 This provides new geometric insight for the entries of the KCF Moreover, as a byproduct, we obtain a decomposition result for linear relations, which completes the considerations in [24 The present paper can be viewed as the link between two different fields in linear algebra: linear relations and matrix pairs (or matrix pencils) The paper is organized as follows: In Section 2 we give a short but self-contained introduction to the theory of linear relations which covers all relevant notions like Jordan chains, singular chains and multishifts In particular, for readers not so familiar with linear relations, we illustrate these concepts with simple examples In Section 3 we show that any linear relation has a representation of the form (13) which is called image and kernel representation, respectively The KCF is recalled in Section 4 and its block entries are related to some properties of linear relations Using the KCF of the image representation of a linear relation, we obtain a full characterization of a linear relation in terms of its Jordan chains at finite eigenvalues, its Jordan chains at infinity, its singular chains and its multi-shift part As a byproduct, a canonical decomposition for linear relations is shown In Section 5 we recall the notion of Wong sequences and exploit them to derive representations for the root and Jordan chain manifolds of a linear relation 2 Preliminaries: Linear relations Let H and be linear spaces A linear relation A in H is a (linear) subspace of H A linear relation A is usually viewed as a multivalued mapping We restrict ourselves to finite dimensional spaces H = C n and = C m Moreover, if H and coincide, ie, if H = = C n, then we briefly say that A is a linear relation in C n instead of C n C n Most of the definitions below remain valid for infinite-dimensional spaces, see eg [13 2

Linear mappings (eg given via a matrix) are always identified with linear relations via their graphs For the general study of linear relations we refer to the monographs [12, 16, see also [1, 26 It is usual to write the elements of A as (x, y) for x C n and y C m In the older literature it is also usual to write the elements of A as column vectors ( x y ) Here, we agree not to distinguish between these two notions By dom A and ran A we denote the domain and the range of a linear relation A in C n C m, dom A = { x C n y C m : (x, y) A } and ran A = { y C m x C n : (x, y) A } Furthermore, ker A and mul A denote the kernel and the multivalued part of A, ker A = { x C n (x, ) A } and mul A = { y C m (, y) A } A linear relation A is the graph of an operator if, and only if, mul A = {} The inverse A 1 is given by A 1 = { (y, x) C m C n (x, y) A } (21) For relations A and B in C n C m the operator-like sum A + B is the relation defined by A + B = { (x, y + z) C n C m (x, y) A, (x, z) B } and for λ C the relation λa is defined by λa = { (x, λy) C n C m (x, y) A }, We illustrate the above definitions by a simple example Example 21 Let e 1, e 2 be the two linearly independent unit vectors in C 2 Define A := span {(, e 1 ), (e 1, e 2 ), (e 2, )}, B := span {(e 2, e 1 ), (e 1, )}, C := span {(e 1, 2e 1 )} and D := span {(e 1, e 2 )}, which are subspaces in C 2 C 2, and hence linear relations in C 2 We have dom A = C 2 dom B = C 2 dom C = span {e 1 } dom D = span {e 1 } ran A = C 2 ran B = span {e 1 } ran C = span {e 1 } ran D = span {e 2 } ker A = span {e 2 } ker B = span {e 1 } ker C = {} ker D = {} mul A = span {e 1 } mul B = {} mul C = {} mul D = {} Moreover, A is not the graph of an operator whereas B is the graph of the linear mapping induced by the matrix [ 1 The relations C and D are the graphs of operators which are defined only on the subset span {e 1 } C 2 and map e 1 to 2e 1, resp e 1 to e 2 The inverses are given by A 1 := span {(, e 2 ), (e 2, e 1 ), (e 1, )}, B 1 := span {(, e 1 )(e 1, e 2 )}, C 1 := span {(e 1, 1 2 e 1)} and D 1 := span {(e 2, e 1 )}, and, the (operator-like) sum of A and C, the sum of C and 5D and the sum of C and D 1 are given by A + C = span {(, e 1 ), (e 1, 2e 1 + e 2 )}, C + 5D = span {(e 1, 2e 1 + 5e 2 )}, and C + D 1 = {} 3

For relations A in C n C m and B in C p C n the product AB is defined as the relation AB = { (x, y) C p C m (x, z) B, (z, y) A for some z C n } (22) We recall the notion of eigenvalues, root manifolds and point spectrum of linear relations Therefore, let A be a linear relation in C n, ie, m = n Then, with the notion of operator-like sum from above, the expression A λ stands for A λi, where I is the identity operator on C n, A λ = { (x, y λx) C n C n (x, y) A } A point λ C is called an eigenvalue of A if ker (A λ) {} and is called an eigenvalue of A if mul A {} The point spectrum σ p (A) is the set of all eigenvalues λ C { } of A The root manifolds R λ (A) and R (A) are defined by R λ (A) := i N ker (A λ) i, R (A) := i N mul A i It is clear that ker (A λ) k ker (A λ) k+1 and mul A k mul A k+1 for any k N By [25, Lemma 34 and the fact that A is a linear relation in a finite dimensional space C n, there exists a natural number n n such that ker (A λ) k = ker (A λ) k+1 for all k n A similar statement holds for mul A k For x ker (A λ) l \ ker (A λ) l 1, l 1, we find (cf (22)) x 1,, x l 1 C n such that or, equivalently, (x, x l 1 ), (x l 1, x l 2 ),, (x 2, x 1 ), (x 1, ) A λ (x, x l 1 + λx), (x l 1, x l 2 + λx l 1 ),, (x 2, x 1 + λx 2 ), (x 1, λx 1 ) A (23) The vectors x 1,, x l 1, x C n are linearly independent and (23) is called a Jordan chain at λ, see [24, Lemma 21 Moreover, we say that it is a Jordan chain of length l Similarly, for y mul A m \ mul A m 1, m 1, there are y 1,, y m 1 C n such that (, y 1 ), (y 1, y 2 ),, (y m 2, y m 1 ), (y m 1, y) A (24) The vectors y 1,, y n 1, y C n are linearly independent and (24) is called a Jordan chain at (cf [24, Lemma 21) Moreover, we say that it is a Jordan chain of length m Obviously, a chain of the form (24) is a Jordan chain at (of length m) if, and only if, is a Jordan chain of A 1 at (of length m) (y, y m 1 ), (y m 1, y m 2 ),, (y 2, y 1 ), (y 1, ) Example 22 For the linear relations B and C from Example 21 we have σ p (B) = {}, and σ p (C) = {2} In addition, for the linear relation B the chain (e 2, e 1 ), (e 1, ) is a Jordan chain of length two at and for C the chain (e 1, 2e 1 ) is a Jordan chain of length one at 2 with R (B) = C 2, and R 2 (C) = span {e 1 } We define the Jordan chain manifold R J (A) as the linear span of all root manifolds, R J (A) := span { R λ (A) λ σ p (A) }, and the finite Jordan chain manifold R f (A) as the linear span of all root manifolds R λ (A) with λ, R f (A) := span { R λ (A) λ C } Obviously, if A is the graph of an operator in the finite dimensional space C n, then R f (A) = R J (A) = C n The converse is not true in general, which is illustrated by the following example 4

Example 23 The linear relation A from Example 21 is not the graph of an operator Its chain (e 1, e 2 ), (e 2, ) is a Jordan chain at and (, e 1 ), (e 1, e 2 ) is a Jordan chain at Therefore, σ p (A) We have R (A) = C 2 and R (A) = C 2, thus Moreover, we obtain R f (A) = R J (A) = C 2 R (A) R (A) {} (25) In the following example we compute the point spectrum of the linear relation A from Example 21 Example 24 From Example 23 we conclude, σ p (A) For λ C \ {} we have (e 1 + λ 1 e 2, λe 1 + e 2 ) = λ(, e 1 ) + (e 1, e 2 ) + λ 1 (e 2, ) A, and, hence, e 1 + λ 1 e 2 ker (A λ) This implies λ σ p (A) and we conclude σ p (A) = C { } It is well-known [24, Proposition 32 and Theorem 44 that it is the property in (25) which is equivalent to σ p (A) = C { } We recall this important fact from [24 in the following lemma Lemma 25 Let A be a linear relation in C n Then σ p (A) = C { } if, and only if, R (A) R (A) {} In the sequel the set R (A) R (A) is useful in the study of linear relations and it is called the singular chain manifold, R c (A) := R (A) R (A) Moreover, A is called completely singular, if A = A (R c (A) R c (A)) (26) For any x R c (A) \ {} there exist linearly independent x 1,, x k C n (see [24, Lemma 31) such that x = x j for some j {1,, k} and (, x 1 ), (x 1, x 2 ),, (x k 1, x k ), (x k, ) A (27) A chain of this form is called a singular chain Moreover, we say that it is a singular chain of length k A linear relation A is completely singular if, and only if, it is the span of singular chains of the form (27), see [24, Section 7 If R c (A) = {} (and hence σ p (A) C { } by Lemma 25), then (see eg [24, Theorem 46) the number of eigenvalues in σ p (A) is bounded by the dimension of the linear subspace A If, in addition, the linear relation A consists only of Jordan chains at the (finitely many) eigenvalues, then we call A a Jordan relation Apart from linear relations in C n with finite point spectrum and with point spectrum equal to C { } there exist also linear relations with no point spectrum Example 26 For the linear relation D from Example 21 we have mul D = {} and hence / σ p (D) Moreover, for λ C, D λ = span {(e 1, e 2 λe 1 )} As e 1, e 2 are linearly independent unit vectors, we have e 2 λe 1 for all λ C, thus λ / σ p (A) which implies σ p (A) = (28) We use property (28) for the definition of a subclass of all linear relations: A linear relation A in C n with σ p (A) = is called a multishift (see, eg, [24, Section 8) 5

3 Image and kernel representations of linear relations In this section we derive two representations for a linear relation A in C n It is well-known that if there exists a complex number µ with ran (A µ) = C n and ker (A µ) = {}, then the inverse of A µ is a linear mapping defined on C n, that is (A µ) 1 can be identified with a matrix in C n n and A admits a representation of the form (see, eg, [14, Proposition 22) A = { ((A µ) 1 x, (I + µ(a µ) 1 )x) C n C n x C n } (31) Definition 31 A representation of a linear relation A in C n of the form { } [ A = (F z, z) C n C n z C d F = ran with d N and matrices F, C n d is called an image representation of A A representation of A of the form A = { (x, y) C n C n Ax = Ey } = ker [A, E with matrices A, E C r n and r N is called a kernel representation of A Observe that (31) is a special case of an image representation where d = n Let us consider a linear relation A in C n with image and kernel representation as in Definition 31 As usual, we identify the matrices F and with the corresponding relations via their graphs, { } { } F = (x, F x) C d C n x C d and = (y, y) C d C n y C d We have, see (21), F 1 = { (F y, y) C n C d y C d } and with (22) F 1 = { (x, y) (x, z) F 1, (z, y) for some z C d } = { (x, z) (z, x) F, (z, z) for some z C d } = { (F z, z) z C d } [ F = ran = A (32) Similarly, if we identify A and E with the corresponding relations A = { (x, Ax) C n C r x C n } and E = { (y, Ey) C n C r y C n }, then, with E 1 = { (Ey, y) C r C n y C n } and with (22), we obtain E 1 A = { (x, y) (x, z) A, (z, y) E 1 for some z C r } We have thus proved the following result = { (x, y) C n C n Ax = Ey } = ker [A, E = A Lemma 32 Let A be a linear relation in C n with image and kernel representation as in Definition 31 Then we have A = F 1 = E 1 A In the following we show that for every relation in C n an image and a kernel representation exist Theorem 33 Let A be a linear relation in C n with dim A = d Then there exist matrices F, C n d with [ F rk = d (34) such that { } A = (F z, z) C n C n z C d = ran 6 (33) [ F = F 1 (35)

Moreover, for r = 2n d there exist matrices A, E C r n with rk [A, E = r such that A = { (x, y) C n C n Ax = Ey } = ker [A, E = E 1 A (36) Proof The third equality in (35) and in (36) follows from Lemma 32 We have A A = C n C n and dim A = r Therefore, we find vector space isomorphisms V : C d A and W : C r A (37) Denote by P 1 and P 2 the orthogonal projection in C n C n onto the first and second component, respectively Then we obtain [ [ P1 V A = ran and A P1 W = ran (38) P 2 V P 2 W and (35) is shown In order to show (36) we continue with A = ( ran [ ) P1 W = { (y, z) C n C n x C r : y P P 2 W 1 W x + z P 2 W x = } = ker [(P 1 W ), (P 2 W ) = ker [W P 1, W P 2 The kernel representation (36) of a linear relation A was already considered in [4, 5 using the notation E\A := { (x, y) C n C n Ax = Ey } However, Lemma 32 and Theorem 33 show E\A = E 1 A Remark 34 It seems natural to single out the cases when A allows an image representation (a kernel representation) with square matrices F and (A and E, respectively) Obviously, if dim A n, then we can choose in (37) a mapping V : C n A such that V is surjective but not necessarily injective Note that in this case V is an isomorphism if, and only if, dim A = n Then we obtain as in (38) an image representation with square matrices Conversely, if A has an image representaion with square matrices F, C n n, then [ F dim A = dim ran n Therefore there exists an image representation of A with square matrices if, and only if, dim A n By similar arguments, there exists a kernel representation of A with square matrices if, and only if, dim A n 4 Kronecker canonical form In this section we recall the KCF and show how it is related to the decomposition of an associated linear relation We introduce the following notation: Let α be a multi-index, α = (α 1,, α l ) N l As usual, the absolute value of α is α = l i=1 α i For k N we define the matrices [ 1 N k = C k k, N α = diag (N α1,, N αl ) C α α 1 7

For k N, k > 1, we define the rectangular matrices [ 1 [ 1 K k =, L k = 1 1 C (k 1) k For some multi-index α = (α 1,, α l ) N l, with α j > 1 for all j = 1,, l, we define K α = diag (K α1,, K αl ), L α = diag (L α1,, L αl ) C ( α l) α (41) We extend the above notion to multi-indices α = (α 1,, α l ) N l where some entries are equal to 1 but α (1, 1,, 1) In this case, we define K α and L α in the following way: Collect in a multiindex α all entries of α which are larger than 1, α = (α j1,, α jk ) with 1 k l 1 We have α k = α l and K α (L α ) is defined via (41) and is of size ( α k) α Then K α (L α, respectively) is defined by augmenting K α (L α, respectively) by α α zero columns without changing the number of rows in such a way that to each entry α j = 1 in α there corresponds a zero column located at the j-th position Then K α and L α are of size ( α l) α As an example consider K (1,2) := [ 1 1 and K (1,2,1,3) := 1 1 Moreover, we extend the above notion to the case α = (1, 1,, 1) N l, if K α and L α are diagonal blocks entries of a larger matrix with other proper defined matrix elements For instance, let M C m n, α = (1, 1,, 1) N l Then we define diag (M, K α ) := [M, m l C m (n+l), diag (K α, M) := [ m l, M C m (n+l), diag (M, L α ) := [M, m l C m (n+l), diag (L α, M) := [ m l, M C m (n+l) Similarly, [ [ M M diag (M, Kα ) := C (m+l) n, diag (M, L α ) := C (m+l) n l n l n Finally, we define for β = (1, 1,, 1) N l and γ = (1, 1,, 1) N m diag (K β, Kγ ) = m l C m l Some of the properties of the matrices K α, L α and N α are collected in the following lemma Lemma 41 For α N l we have rk K α = rk L α = rk N α = α l Furthermore, for all λ C, and in particular rk(λk α L α ) = α l, ker K α = ker (λk α L α ) = {} Kronecker proved in [17 that any pair of matrices F, can be transformed into a canonical form, see also [8, 9, 15 Here we refer to the version in [15 8

Theorem 42 (Kronecker canonical form) For any pair of matrices F, C n d there exist invertible matrices W C n n and T C d d such that I n A W F T = and W T = (42) N α K β Kγ I α L β L γ for some A C n n in Jordan canonical form and multi-indices α N nα, β N n β, γ N nγ The multi-indices α, β, γ are unique up to a permutation of their respective entries Further, the matrix A is unique up to a permutation of its Jordan blocks The entries of the multi-indices α, β, γ are called minimal indices and elementary divisors and play an important role in the analysis of matrix pairs (F, ), see eg [8, 9, 21, 22, 23, where the entries of α are the orders of the infinite elementary divisors, the entries of β are the column minimal indices and the entries of γ are the row minimal indices In what follows, we investigate the relationship between a linear relation A and the KCF of the matrices F and from its image representation (cf Definition 31) Then with the notation from Theorem 42 we find A = ran [ F = ran I n N α K β Kγ (43) A I α L β L γ In the following proposition we collect some properties of the multi-index β from Theorem 42 and obtain a characterization of the dimension of A in terms of the indices which appear in the KCF (42) Proposition 43 Let A be a linear relation in C n with dim A = d 1 Let F, C n d with rk [ [ F = d be such that A = ran F and let W C n n, T C d d be invertible matrices such that W F T and W T are in KCF (42) Then the following statements hold (i) Either n β = or β i 2 for all i = 1,, n β (ii) The dimension of A satisfies dim A = n + α + β + γ n γ (iii) We have dim A n if, and only if, n β n γ (iv) We have dim A n if, and only if, n β n γ Proof Since the entries of the multi-index β which equal 1 correspond to zero columns in [ F, statement (i) follows from rk [ F = d In order to show (ii), observe that for k > 1 [ Kk L k C 2(k 1) k and [ K k L k C 2k (k 1) with rk [ Kk L k [ K k = k and rk = k 1 L k 9

Therefore, for β N n β and γ N nγ and with (i) we see [ [ Kβ K γ rk = β and rk = γ n γ L β Then (ii) follows from (43) As n is the number of rows in the KCF (42), and a comparison with (ii) yields (iii) and (iv) L γ n = n + α + β n β + γ The properties of the linear relation A are encoded in the different blocks of the KCF We start with the following simple example Example 44 Let A be a linear relation in C n in the form (43) Assume, for simplicity, that W and T in (42) are equal to the identity map (i) Assume that in (42) only the first row appears, ie the matrices F and are of the form F = I n and = A, for some A C n n in Jordan canonical form Then A is given by [ [ { [ F In x A = ran = ran = x C n A x A } For λ C and x C n \ {} we have A x = λx if, and only if, (x, λx) A This is equivalent to (x, ) A λ, hence σ p (A) = σ(a ), where σ(a ) denotes the spectrum of the matrix A consists of finitely many points In particular, the point spectrum of A (ii) Assume that in (42) only the second row appears, ie, the matrices F and are of the form and we have A = ran F = N α and = I α [ F = ran [ Nα I α = { [ } Nα y y C α y The matrix N α has only the eigenvalue zero Hence [ { [ } A 1 I α y = ran = y C α N α y N α has only the eigenvalue zero and A is a Jordan relation with only eigenvalue (iii) Assume that in (42) only the third row appears and that the multi-index β consists of one entry only, β = k for some k N, k > 1, F = K k and = L k Then A = ran [ F = ran [ Kk L k = { [ } Kk x x C k L k x 1

For standard unit vectors e i C k we calculate K k e k = = L k e 1 and for i = 1,, k 1 we denote the unit vectors in C k 1 by ê i and obtain Therefore, which is a singular chain in A K k e i = ê i, L k e i+1 = ê i (, ê k 1 ) = (K k e k, L k e k ) A, (ê k 1, ê k 2 ) = (K k e k 1, L k e k 1 ) A, (ê 2, ê 1 ) = (K k e 2, L k e 2 ) A, (ê 1, ) = (K k e 1, L k e 1 ) A, (iv) Finally, assume that in (42) only the fourth row appears and that the multi-index γ consists of one entry only, γ = k for some k N, k > 1, F = K k and = L k Then A = ran [ [ F K k = ran = L k { [ K k y L k y y Ck 1 } For λ C we have (x, λx) A with x if, and only if, there exists y C k 1 \ {} with x = Kk y and λx = L k y This is equivalent to ker (λk k L k ) {} But Lemma 41 implies ker (λkk L k ) = {} for all λ C, thus ker (A λ) = {} Similarly, x mul A if, and only if, there exists y C k 1 with = Kk y and x = L k y But Lemma 41 implies ker Kk = {} and hence mul A = {} Therefore, σ p(a) = and A is a multishift Example 44 indicates that in the first block of the KCF the (finite) eigenvalues of A = ran [ F are encoded The second block represents the eigenvalue, the third block the singular chains and the fourth the multishifts This relationship is exploited in all details (ie, with emphasis on the number and length of the chains of different types) in the next theorem which is the main result of this paper In what follows two chains are called linearly independent if their entries are linearly independent Theorem 45 Let A be a linear relation in C n with dim A = d 1 Let F, C n d with rk [ F = d be such that A = ran [ F and let W C n n, T C d d be invertible matrices such that W F T and W T are in KCF (42) with α = (α 1,, α nα ), β = (β 1,, β nβ ), and γ = (γ 1,, γ nγ ) (i) For the singular chain manifold we have R c (A) = ( {} n {} α C β n β {} γ ) Moreover, A ( R c (A) R c (A) ) is spanned by n β linearly independent singular chains of lengths β 1 1, β 2 1,, β nβ 1 11

(ii) For the root manifold at we have R (A) = ( {} n C α C β n β {} γ ) Moreover, A ( R (A) R (A) ) is spanned by n α + n β linearly independent chains with n β singular chains of lengths β 1 1, β 2 1,, β nβ 1 and n α Jordan chains at of lengths α 1, α 2,, α nα (iii) For the finite Jordan chain manifold we have R f (A) = ( C n {} α C β n β {} γ ) Moreover, A ( R f (A) R f (A) ) is spanned by a set of linearly independent chains consisting of n β singular chains of lengths β 1 1, β 2 1,, β nβ 1 and the Jordan chains constituted by the Jordan chain vectors of the matrix A In particular, we have σ(a ) σ p (A) (iv) For the Jordan chain manifold we have R J (A) = ( C n C α C β n β {} γ ) Moreover, A ( R f (A) R f (A) ) is spanned by a set of linearly independent chains consisting of n β singular chains of lengths β 1 1, β 2 1,, β nβ 1 and n α Jordan chains at of lengths α 1, α 2,, α nα and the Jordan chains constituted by the Jordan chain vectors of the matrix A Proof Step 1 We show (i) Let x R c (A) \ {} Then there exists a singular chain of the form (27) with linearly independent x 1,, x k C n and x = x j for some j {1,, k} By A = ran [ F there exist z 1,, z k+1 C d such that (, x 1 ) = (F z 1, z 1 ), (x 1, x 2 ) = (F z 2, z 2 ), (44) (x k 1, x k ) = (F z k, z k ), (x k, ) = (F z k+1, z k+1 ) For i {1,, k + 1} define y i = T 1 z i Partitioning y i = (yi,1,, y i,4 ) with y i,1 C n, y i,2 C α, y i,3 C β, y i,4 C γ nγ according to the decomposition (42), we obtain from the first equation in (44) that I n y 1,1 = F z 1 = (W F T )T 1 z 1 = N α y 1,2 K β y 1,3 Kγ y 1,4 and hence y 1,1 = and K γ y 1,4 =, thus, by Lemma 41, y 1,4 = Furthermore, x 1 = z 1 = (W T )T 1 z 1 A y 1,1 = I α y 1,2 L β y 1,3 = L γ y 1,4 y 1,2 L β y 1,3 (45) 12

The second equation in (44) gives I n y 2,1 x 1 = F z 2 = (W F T )T 1 z 2 = N α y 2,2 K β y 2,3 Kγ y 2,4 and a comparison with (45) yields y 2,1 = and Kγ y 2,4 =, thus, by Lemma 41, y 2,4 = Furthermore, x 2 = z 2 = (W T )T 1 z 2 Proceeding in this way, we see A = I α L β L γ x i = y i,2 L β y i,3 y 2,1 y 2,2 y 2,3 y 2,4 = y 2,2 L β y 2,3, i = 1,, k (46) From the last equation in (44) we conclude that A y k+1,1 = z k+1 = (W T )T 1 z k+1 = I α y k+1,2 L β y k+1,3 L γ y k+1,4 and hence y k+1,2 = and L γ y k+1,4 =, thus, by Lemma 41, y k+1,4 = Furthermore, Proceeding in this way gives x k = F z k+1 = (W F T )T 1 z k+1 I n y k+1,1 = N α y k+1,2 K β y k+1,3 = Kγ y k+1,4 and together with (46) we find x i = for all i = 1,, k This implies that x = x j = L β y j,3 y i,2 L β y i,3 y i+1,1 K β y i+1,3, i = 1,, k, = x i = y i+1,1 K β y i+1,3 y k+1,1 K β y k+1,3 ( {} n {} α C β n β {} γ ) 13

Conversely, let us start with the case that K β and L β consist of one block only That is, n β = 1 and β = k with some k N, k 2, cf Proposition 43 Define y 1 := 1, y 2 := 1,, y k := 1 C k (47) Then = K k y 1, L k y k =, and L k y 1 = K k y 2 = 1 Ck 1,, L k y k 1 = K k y k = 1 Ck 1 (48) Set x :=, x i := (,, (L k y i ), ), for i = 1,, k 1, and x k := As W F T, W T are in KCF (42), we find that, invoking (48), ( xi 1 x i ) = I n N α K k K γ A I α L k L γ y i = [ F T y i ran [ F = A for i = 1,, k, and hence (, x 1 ), (x 1, x 2 ),, (x k 2, x k 1 ), (x k 1, ) A (49) We see from (48) that L k y 1,, L k y k 1 are the unit vectors in C k 1 Hence, x 1,, x k 1 are linearly independent and (49) constitutes a singular chain as in (27) Therefore, if the multi-index β has only one entry and if this entry equals k, then (49) is a singular chain of length k 1 and in particular R c (A) = ( {} n {} α C k 1 {} γ ) This, (43), and the fact that [y k,, y 1 = I k C k k yield A ( R c (A) R c (A) ) = ran K k L k [y k,, y 1 = ran [L k y k 1,, L k y 1, [, L k y k 1,, L k y 1 = span {(, x 1 ), (x 1, x 2 ),, (x k 2, x k 1 ), (x k 1, )} 14

In the general case β N n β with n β > 1 we have n β decoupled blocks in K β and L β and for each block the above construction leads to a singular chain of A In this manner we obtain n β linearly independent singular chains of lengths β 1 1, β 2 1,, β nβ 1, resp, which span A ( R c (A) R c (A) ) Step 2 We show (ii) Let y R (A) Then there exists a Jordan chain at of the form (24) with linearly independent y 1,, y m 1, y C n Set y m := y As in the proof of (46) it follows that for some z 1,2,, z m,2 C α, z 1,3,, z m,3 C β we have y i = z i,2 L β z i,3, i = 1,, m This proves y ( {} n C α C β n β {} γ ) Conversely, let z 2 C α, z 3 C β n β and set x := (, z 2, z 3, ) It follows from (i) that (,, z 3, ) R c (A) R (A) Assume that N α consists of one block only That is, n α = 1 and α = k for some k 1 Choose y 1,, y k as in (47) Then = N k y 1, y 1 = N k y 2,, y k 1 = N k y k (41) Set x :=, x i := (, y i,, ) for i = 1,, k, we obtain with (42) and (41), ( xi 1 x i ) = for i = 1,, k and therefore I n N k K β Kγ A I k L β L γ [ [ y i F = T y i F ran = A (, x 1 ), (x 1, x 2 ),, (x k 2, x k 1 ), (x k 1, x k ) A (411) The vectors y 1,, y k in (47) are linearly independent and then the same holds for x 1,, x k Thus, (411) constitutes a Jordan chain of A at of length k In particular, since z 2 span {y 1,, y k } it follows that x R c (A) + span {x 1,, x k } and thus we have shown that R (A) = ( {} n C k C β n β {} γ ) 15

This and the first step of this proof yield A ( R (A) R (A) ) = ran = A ( R c (A) R c (A) ) ran N k K β I k L β N k [y k,, y 1, I k where we used that [y k,, y 1 = I k C k k and A B denotes the direct sum of two subspaces A and B with A B = {} Therefore A ( R (A) R (A) ) = A ( R c (A) R c (A) ) ran [y k 1,, y 1, [y k,, y 2, y 1 = A ( R c (A) R c (A) ) span {(, x 1 ), (x 1, x 2 ),, (x k 1, x k )}, Hence, (ii) is proved in the case n α = 1 and α = k In general, if α N nα, then there are n α decoupled blocks in N α For each block the above construction leads to a Jordan chain of A at and we obtain n α linearly independent Jordan chains at of lengths α 1,, α nα, respectively, which lead, together with the singular chains, to the span of A ( R (A) R (A) ) Step 3 We show (iii) Let x R f (A) Since R f (A) has a finite basis, there exist k N and pairwise distinct λ 1,, λ k C such that k x R λi (A) Therefore, we find i=1 v i ker (A λ i ) n, i = 1,, k, 16

such that x = v 1 + + v k We show that v i ( C n {} α C β n β {} γ ) for all i = 1,, k For simplicity, let v := v i and λ := λ i for some i {1,, k} By (23) there exist v 1,, v n 1 C n such that (v, v n 1 + λv), (v n 1, v n 2 + λv n 1 ),, (v 2, v 1 + λv 2 ), (v 1, λv 1 ) A, where, for simplicity, we do not assume that the vectors v 1,, v n 1, v are linearly independent Set v n := v We obtain from A = ran [ F the existence of z1,, z n C d such that (v n, v n 1 + λv n ) = (F z n, z n ), (v n 1, v n 2 + λv n 1 ) = (F z n 1, z n 1 ), (v 2, v 1 + λv 2 ) = (F z 2, z 2 ), (412) (v 1, λv 1 ) = (F z 1, z 1 ) (413) Define y i := T 1 z i for i = 1,, n Partitioning y i = (yi,1,, y i,4 ) with y i,1 C n, y i,2 C α, y i,3 C β, y i,4 C γ nγ according to the decomposition (42), we obtain from (413) I n y 1,1 v 1 = F z 1 = (W F T )T 1 z 1 = N α y 1,2 K β y 1,3 Kγ y 1,4 and Therefore, A y 1,1 λv 1 = z 1 = (W T )T 1 z 1 = I α y 1,2 L β y 1,3 L γ y 1,4 (λn α I α )y 1,2 = and (λk γ L γ )y 1,4 =, thus, by invertibility of λn α I α and Lemma 41, y 1,2 = and y 1,4 = Similarly (412) gives (λn α I α )y 2,2 = N α y 1,2 = and (λk γ L γ )y 2,4 = K γ y 1,4 =, and hence y 2,2 = and y 2,4 = Solving the remaining equations successively, we obtain finally y n,2 = and y n,4 =, which implies y n,1 v = v n = F z n = K β y n,3 and R f (A) ( C n {} α C β n β {} γ ) follows We prove the converse inclusion From (i) it follows ( C n {} α C β n β {} γ ) ( {} n {} α C β n β {} γ ) = R c (A) R (A) R f (A) We show ( C n {} α {} β n β {} γ ) R f (A) The space C n has a basis consisting of Jordan chains of the matrix A Let v 1,, v k be a Jordan chain of A at λ C of length k Then v 1,, v k are linearly independent and satisfy (A λ)v k = v k 1, (A λ)v k 1 = v k 2,, (A λ)v 2 = v 1, (A λ)v 1 = (414) 17

Set Then we obtain and we see with (414) that for j = 2,, k, x j := ((v j ),,, ) for j = 1,, k I n v j v j x j = N α K β = F T Kγ λv j + v j 1 A v j λx j + x j 1 = = A v j v j = I α L β = T L γ and for j = 1 λv 1 A v 1 λx 1 = = A v 1 v 1 = I α L β = T L γ Therefore, for j = 2,, k, ( x j ) λx j + x j 1 ran [ F and ( x1 ) ran λx 1 [ F, hence (x k, λx k + x k 1 ), (x k 1, λx k 1 + x k 2 ),, (x 2, λx 2 + x 1 ), (x 1, λx 1 ) A (415) The vectors x 1,, x k are linear independent because W is invertible and the vectors v 1,, v k are linear independent Therefore, (415) constitutes a Jordan chain of A at λ of length k Since the Jordan chain v 1,, v k was arbitrary we have shown that R f (A) = ( C n {} α C β n β {} γ ) The remaining statements of (iii) follow from (i), the construction of (415) above and the observation 18

that A ( R f (A) R f (A) ) I n K β = ran A L β = A ( R c (A) R c (A) ) ran I n A In particular, we see that σ(a ) σ p (A) Step 4 We show (iv) Since R J (A) = R (A) + R f (A) it follows from (i) and (iii) that ( R J (A) = C n C α C β n β {} γ ) The remaining statements of (iv) follow from A ( R J (A) R J (A) ) I n N α K β = ran = A ( R f (A) R f (A) ) ran A I α L β I n A and the same arguments for the Jordan chains of A as in Step 3 Corollary 46 With the notation from Theorem 45 we have n = dim R f (A) dim R c (A), α = dim R (A) dim R c (A), β n β = dim R c (A), γ = n dim R J (A) Corollary 47 With the notation from Theorem 45 we have C { }, if n β, σ(a ) { }, if n β = and n α σ p (A) = σ(a ), if n β = n α =, if n β = n α = n = (416) 19

Proof First observe that by Lemma 25, σ p (A) = C { } if, and only if, R c (A) {} By Theorem 45 and Proposition 43, this is equivalent to n β and the first line in (416) is shown For the rest of the proof we assume n β = A complex number λ C belongs to σ p (A) if, and only if, there exists x C n \{} with (x, λx) A Equivalently, by Theorem 45, there exist y 1 C n, y 2 C α and y 4 C γ nγ such that at least one of them is non-zero, with y 1 x = N α y 2 and λx = Kγ y 4 A y 1 y 2 L γ y 4 With Lemma 41 we see that this is equivalent to y 2 = y 4 =, y 1 and A y 1 = λy 1, or, what is the same, λ σ(a ) Hence, in the case n β =, we have σ p (A) \ { } = σ(a ) (417) It remains to consider the point By definition, σ p (A) if, and only if, R (A) {} which is, by Theorem 45, equivalent to n α This and (417) show the second and third line in (416), whereas the last line in (416) is now obvious Using Theorem 45 we may derive a characterization for A being completely singular, a Jordan relation or a multishift Proposition 48 With the notation from Theorem 45 we have that the linear relation A is (i) completely singular (see (26)) if, and only if, n = n α = and γ = (1,, 1); (ii) a Jordan relation if, and only if, n β = and γ = (1,, 1); (iii) a multishift if, and only if, n = n α = n β = Proof We show (i) By (26) A is completely singular if, and only if, A = A (R c (A) R c (A)) Invoking Theorem 45 and (43) this is equivalent to I n N α K β Kγ K β ran = ran A I α L β L β L γ This is true if, and only if, n = n α = and γ = (1,, 1) We show (ii) By definition (cf Section 2), A is a Jordan relation if, and only if, R c (A) = {} and A = A (R J (A) R J (A)) Invoking Theorem 45 this is equivalent to n β = and I n I n N α N α Kγ ran = ran A A I α I α L γ 2

Similar to (i), this is true if, and only if, n β = and γ = (1,, 1), thus (ii) follows We show (iii) By definition, A is a multishift if, and only if, σ p (A) = Then (iii) follows from Corollary 47 The KCF leads to a decomposition of a linear relation A in a natural way For this we introduce the following notion Here stands for the direct sum of subspaces Definition 49 Let A be a linear relation in C n A decomposition A = A 1 A k, k N, of a linear relation A is called completely reduced, if there exist subspaces V 1,, V k of C n such that (i) V 1 V k = C n and (ii) A j = A (V j V j ) for all j = 1,, k In [24 it is shown that any linear relation can be decomposed into a direct sum of a Jordan relation, a completely singular relation and a multishift However, this decomposition is in general not completely reduced The KCF resolves this problem Proposition 41 With the notation from Theorem 45 define A S := ran A M := ran K β, A J := ran L β K γ L γ I n N α A I α, Then A S is completely singular, A J is a Jordan relation, A M is a multishift and A = A S A J A M is a completely reduced decomposition Proof We show that the decomposition is completely reduced Set ( V 1 := R c (A) = {} n {} α C β n β {} γ ), ( V 2 := C n C α {} β n β {} γ ), ( V 3 := {} n {} α {} β n β C γ ) 21

Then it is clear that V 1 V 2 V 3 = C n and A S = A (V 1 V 1 ), A J = A (V 2 V 2 ), A M = A (V 3 V 3 ) It is clear that R c (A S ) = R c (A) = V 1 and hence A S is completely singular Furthermore, R c (A J ) = {} and R J (A J ) = V 2, thus A J is a Jordan relation Finally, it follows from Corollary 47 that σ p (A M ) = and hence A M is a multishift 5 Wong sequences Recently, see [7, 8, 9, Wong sequences are used to prove the KCF and compared to the proof by antmacher [15 they provide some geometrical insight The Wong sequences have their origin in Wong [27 Wong sequences are also useful to describe the different parts of a linear relation In this section we derive representations for the root and Jordan chain manifolds of a linear relation in terms of the Wong sequences For E, A C r n the Wong sequences are defined as the sequences of subspaces (V i ) and (W i ), V = C n, V i+1 = { x C n Ax EV i }, i N, W = {}, W i+1 = { x C n Ex AW i }, i N The limits of the Wong sequences are denoted by V = i N V i and W = i N W i For F, C n d alternative Wong sequences are defined by the sequences ( V i ) and (Ŵi), V = C n, { Vi+1 = F x x C d, x V } i, i N, Ŵ = {}, { } Ŵ i+1 = x x C d, F x Ŵi, i N, with corresponding limits V = Vi and Ŵ = Ŵ i i N i N Theorem 51 Let A be a linear relation in C n with dim A = d and let A, E C r n, r = 2n d, with rk [A, E = r and F, C n d with rk [ F = d be such that A = ker [A, E = ran [ F Then we have V W = V Ŵ = R c (A), V = V = R f (A) = dom A n, W = Ŵ = R (A), V + W = V + Ŵ = R J (A) Proof We show that, for all i N, dom A i = V i = V i, (51) mul A i = W i = Ŵi (52) 22

We prove the first equality in (51) by induction For i = the statement is true, so assume that it holds for some i N Then dom A i+1 = { x C n y C n : (x, y) A = ker [A, E and y dom A i } = { x C n y C n : Ax = Ey and y V i } = V i+1 The second equality in (51) follows from { dom A i+1 = x C n y C n : (x, y) A = ran [ F and y dom A i = V } i { = x C n y C n z C d : x = F z and y = z and y V } i { = x C n z C d : x = F z and z V } i = V i+1 The proof of (52) is analogous and omitted From (51) and (52) and the fact that, by finite dimensionality, V = V n, V = V n, W = W n, Ŵ = Ŵn it now follows that dom A n = V = V and R (A) = W = Ŵ Next we show that V = R f (A) To this end, let W C n n, T C d d be invertible matrices such that W F T and W T are in KCF (42) We prove that i N : Vi = ( C n ran N i α C β n β ran(n γ ) i) For i = the statement is true Assume that the statement is true for some i N We obtain { V i+1 = F x x C d, x V } i = F T y A I α L β y V i L γ Now, by assumption, V i = ( C n ran N i α C β n β ran(n γ ) i) and we have I n y 1 V i+1 = N α y 2 K β y 3 Kγ y 4 y 1 C n, y 2 ran N i α, y 3 C β, L γ y 4 ran(n γ ) i = ( C n ran N i+1 α C β n β ran(n γ ) i+1), where for the last equality we need to show that { K γ y 4 L γ y 4 ran(n γ ) i } = ran(n γ ) i+1 Observe that N γ L γ = K γ and N γ K γ = K γ N γ 1, where γ 1 = (γ 1 1,, γ nγ 1) and if γ j = 1 for some j {1,, n γ }, then we define N γ 1 = diag (N γ1,, N γj 1, N γj +1,, N γnγ ) Furthermore, we have that for any j, k N, k 2, and v C k 1, v = (v 1,, v k 1 ), L k v ran(n k )j v k j = = v k 1 = v ran(n k 1 )j, 23

from which it follows that { v C γ nγ L γ v ran(n γ ) j } = ran(n γ 1) j Now we are in the position to infer that ran(n γ ) i+1 = ran(n γ ) i+1 L γ = ran(n γ ) i K γ { = ran Kγ (Nγ 1) i = Kγ v L γ v ran(n γ ) i } Since N n α = and (N γ ) n = it now follows with Theorem 45 that dom A n = V = ( C n {} α C β n β {} γ ) = R f (A) The remaining statements can now be concluded from Theorem 45 Equations (51) and (52) provide a direct connection between the Wong sequences and the corresponding linear relation A In [8 it is shown how the Wong sequences can be used to define a basis transformation which puts the matrix pair (F, ), where A = ran [ F, into KCF Moreover, in [8, 9 it is shown that certain Wong sequences completely determine the KCF In this sense, the KCF of F and is completely determined by the linear relation A = ran [ F 6 Conclusion We have shown how the Kronecker canonical form provides a natural decomposition of a linear relation into a completely singular relation, a Jordan relation and a multishift Furthermore, the KCF can be used for a complete description of the structure of a linear relation up to the structure of the multishift part In this sense we have derived an analogue to the Jordan canonical form for matrices; this is new for linear relations On the other hand, the KCF of a given matrix pair is completely described by the corresponding linear relation, which can be established using Wong sequences The KCF is widely used in the study of matrix pairs or, what is the same, in the investigation of DAEs A famous unsolved problem in the theory of matrix pairs is the distance to the nearest singular pair, see [11 It is possible to characterize regularity and singularity in terms of the induced linear relation For linear relations, the effect of perturbations can be studied using the gap metric, see [3, 19, or by utilizing results for finite dimensional perturbations, see eg [2 In future research one may use the deep connection between linear relations and matrix pencils presented in the present paper In particular, existing perturbation theory for linear relations is now available for the study of matrix pencils and DAEs References [1 R Arens Operational calculus of linear relations Pacific J Math 11:9 23, 1961 [2 T Ya Azizov, J Behrndt, P Jonas, and C Trunk Compact and finite rank perturbations of linear relations in Hilbert spaces Integral Equations Operator Theory, 63:151-163, 29 [3 T Ya Azizov, J Behrndt, P Jonas, and C Trunk Spectral points of type π + and type π for closed linear relations in Krein spaces J London Math Soc, 83:768 788, 211 [4 P Benner and R Byers Evaluating products of matrix pencils and collapsing matrix products Numer Linear Algebra Appl, 8:357 38, 21 24

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