- Email: [email protected]
GRASSMANN MATRICES, DETERMINANTAL ASSIGNMENT PROBLEM AND APPROXIMATE DECOMPOSABILITY
GRASSMANN MATRICES, DETERMINANTAL ASSIGNMENT PROBLEM AND APPROXIMATE DECOMPOSABILITY N. Karcanias ∗ J. Leventides ∗∗ ∗
Control Engineering Centre, School of Engineering and Mathematical Sciences, City University Northampton Square, London EC1V 0HB, UK [email protected] ∗∗ Department of Economics, Section of Mathematics and Informatics, University of Athens Pezmazoglou 8, Athens, Greece [email protected]
Vm Abstract: The exterior equation v 1 ∧ v 2 ∧ . . . ∧ v m = z, z ∈ U, v i ∈ U, U an n−dimensional vector space over F, is an integral part of the study of the Determinantal Assignment Problem (DAP) of linear systems and its solvability (decomposability of z) is characterised by the Quadratic Pl¨ ucker Relations (QPR). An alternative new test for decomposability of z is given, in terms of the rank properties of V the Grassmann matrix, Φm n (z), which is constructed by the m coordinates of z ∈ U. It is shown that the exterior equation is solvable (z is decomposable), if and only if dim Nnm (z) = m, where Nnm (z) , Nr {Φm n (z)}. If z ˜ is simply defined by is decomposable, then the solution space Vz = sp{v i , i ∈ m} Nnm (z). The linear algebra formulation of the decomposability problem provides an alternative framework (to that defined by the QPRs) for the study of solvability and computation of solutions of DAP and enables the definition and study of “approximate solutions” of exterior equations as a distance problem. For the case of m = 2, n = 4 a solution to approximate decomposability is given and its properties c are linked to the singular values of Φ24 (z). Copyright °2002 IFAC Keywords: Frequency Assignment Problems, Linear Algebra, Decomposability of Multivectors
1. INTRODUCTION The Determinantal Assignment Problem (DAP) has emerged as the abstract problem to which the study of pole, zero assignment of linear systems may be reduced (Karcanias and Giannakopoulos, 1984), (Leventides and Karcanias, 1995a). The multilinear nature of DAP suggests that the natural framework for its study is that of exterior algebra (Marcus, 1973). The study of DAP (Karcanias and Giannakopoulos, 1984) may be reduced to a
linear problem of zero assignment of polynomial combinants and a standard problem of multilinear algebra, that is the decomposability of multivectors (Marcus, 1973). The solution of the linear subproblem, whenever it exists, defines a linear space in a projective space P τ , whereas decomposability is characterised by the set of Quadratic Pl¨ ucker Relations (QPR), which define the Grassmann variety of P τ (Hodge and Pedoe, 1952), (Griffiths and Harris, 1978). Thus, solvability of
DAP is reduced to a problem of finding real intersections between the linear variety and the Grassmann variety of P τ . The importance of tools and techniques of algebraic geometry for control theory problems has been demonstrated by the work in (Martin and Hermann, 1978), (Brockett and Byrnes, 1981), (Leventides and Karcanias, 1995a), (Leventides and Karcanias, 1995b) etc. The approach adopted in (Karcanias and Giannakopoulos, 1984), (Karcanias and Giannakopoulos, 1989) differs from that in (Martin and Hermann, 1978), (Brockett and Byrnes, 1981) in the sense that the problem is studied in a projective, rather than an affine space setting; the former approach relies on exterior algebra and on the explicit description of the Grassmann variety, in terms of the QPRs, and has the advantage of being computational. The multilinear nature of DAP has been recently handled by a ”blow up” type methodology, using the notion of degenerate solution and known as ”Global Linearisation” (Leventides and Karcanias, 1995a). The aim of this paper is to give an alternative, linear algebra type, criterion for decomposability of multivectors to that defined by the QPRs and then introduce the notion of “approximate decomposability” that is an integral part of the effort of modifying suitably the overall framework to be able to handle issues of uncertainty. The problem Vm of decomposability of the multivector z ∈ U, where U is a vector space, is equivalent to the solvability of the exterior equation v 1 ∧ v 2 ∧ . . . ∧ v m = z, with v i ∈ U. The conditions for decomposability are given by the set of QPRs (Marcus, 1973) and the solution space Vz = sp{v i , i ∈ m} ˜ may be constructed as shown in (Giannakopoulos et al., 1983). The present approach handles simultaneously the question of decomposabilityV and m the reconstruction of Vz . For every z ∈ U with coordinates {aω , ω ∈ Qm,n }, the Grassmann matrix Φm n (z) of z is defined. It is shown, that rank{Φm (z)} ≥ n − m, for all z 6= 0, and z is n decomposable, if and only if, the equality sign holds. If rank{Φm n (z)} = n − m, then the solution space Vz is defined by Vz , Nr {Φm n (z)}. The rank based test for decomposability is easier to handle than the QPRs and provides a simple method for the computation of Vz . Furthermore, the matrix version formulation of the decomposability allows the definition of the notion of “approximate decomposability” that is studied for the one QPR case in the last section. Throughout the paper the following notation is adopted: If F is a field, or ring then F m×n denotes the set of m × n matrices over F. If H is a map, then R(H), Nr (H), Nl (H) denote the range, right, left nullspaces respectively. Qk,n denotes the set of lexicographically ordered, strictly increasing sequences of k integers from the set
n ˜ , {1, 2, . . . , n}. If V is a vector space and {v i1 , . . . , v ik } are vectors of V, then v i1 ∧ . . . ∧ v ik = v ω ∧, ω V = (i1 , . . . , ik ) denotes their exterior r product and V the r−th exterior power of V (Marcus, 1973). If H ∈ F m×n and r ≤ min{m, n}, then Cr (H) denotes the r−th compound matrix of H (Marcus, 1973). The proof of the results is given in (Karcanias and Leventides, 2007).
2. THE GENERAL DETERMINANTAL ASSIGNMENT PROBLEM Let M (s) ∈ Rm×l [s], m > l, rankR[s] {M (s)} = l and let H , {H(s) : H(s) ∈ Rl×m [s], rankR[s] {H(s)} = l}; the subset of H defined by all H ∈ Rl×m will be denoted by HR . Finding H ∈ H such that the polynomial fM (s, H) = det{H(s)M (s)}
(1)
has assigned zeros, is defined as the Determinantal Assignment Problem (DAP); if H ∈ HR , then the corresponding problem is defined as the constant DAP (R−DAP) (Karcanias and Giannakopoulos, 1984). The different versions of DAP have been introduced as the abstract unifying descriptions of frequency assignment problems (pole,zero) that arise in linear systems theory. The general case, DAP, covers the dynamic version of frequency assignment problems. If we require that fM (s, H) is an arbitrary Hurwitz polynomial, then different classes of Determinantal Stabilisation Problems (DSP) are defined. Let hti , mi , i ∈ ˜l be the rows of H ∈ H, columns of M (s). Then, (H) = ht1 ∧ . . . ∧ htl = ht ∧ ∈ µ Cl¶ m R1×q [s], q = , Cl (M (s)) = m1 (s) ∧ . . . ∧ l q ml (s) = m(s)∧ ∈ R [s] and by the Binet-Cauchy Theorem (Marcus and Minc, 1964) we have fM (s, H) = Cl (H)Cl (M (s)) =< h∧, m(s)∧ > and thus X fM (s, H) = hω mω (s) (2) ω∈Qm,n
where < ·, · > denotes scalar product, ω = (i1 , . . . , il ) ∈ Ql,m and hω , mω (s) are the entries in h∧, m(s)∧ respectively. Note that hω is the l ×l minor of H, which corresponds to the ω set of rows of H and thus is a multilinear alternating function of the hij entries of H. DAP may be reduced to a linear and a standard multilinear subproblem (Karcanias and Giannakopoulos, 1984): (i) Linear Subproblem of DAP: Let m(s)∧ = p(s) ∈ Rq [s]. Investigate the existence of k(s) ∈ Rq [s] such that for some given α(s) ∈ R[s], d = degα(s), and ed (s) = [1, s, . . . , sd ]t . fp (s, k) = k(s)t p(s) =
=
q X
ki (s)pi (s) = α(s) = αt ed (s)
(3)
i=1
(ii) Multilinear subproblem of DAP: Assume that for the given α(s) part (i) is solvable and let K(α) be the family of solutions. Determine whether there exists H ∈ H, H t = [h1 , . . . , hl ] such that h1 ∧ . . . ∧ hl = k, k ∈ K(α)
(4) ¦
fp (s, k), is defined for a given p(s), and it is called an R[s]−polynomial combinant, if ki ∈ R[s], and as R−polynomial combinant, if ki ∈ R. The solution of the exterior equation is a standard problem of exterior algebra, known as decomposability of multivectors (Marcus, 1973). The solvability of the linear subproblem is a standard problem of linear algebra. The solvability of (4) is characterised by the set of Quadratic Pl¨ ucker Relations (QPR) (Marcus, 1973), which in turn describe the Grassmann variety Ω(l, m) of Pq−1 (Hodge and Pedoe, 1952). Thus, solvability of R−DAP is equivalent to finding real intersections between Ψ(α) and Ω(l, m). The aim of this paper is to provide alternative criteria for solvability of (4), to those defined by the QPRs and introducing the notion of approximate solution of (4). A summary of key notions and results from exterior algebra are summarised first.
3. DECOMPOSABILITY OF MULTIVECTORS: BACKGROUND RESULTS Let U be a vector space over a field F and let G(m, U) be the Grassmannian (set of all m−dimensional subspaces of VpU). For Vp every VVp ∈ G(m, U) the injection map f : VV → U m V is a 1is well defined and if p = m, then Vm dimensional subspace U; if {v i , i ∈ m} ˜ is a Vm of is spanned by v 1 ∧. . .∧v m . Let basis of V, then BU = {ui , i ∈ n ˜ } be a basis of U and BUm = {uω ∧ : uω ∧ = ui1 ∧ . . . ∧ uim , ω = (i1 , . .V . , im ) ∈ Qm,n } m denote the corresponding basis of U. The genVm U may be expressed as eral vector z ∈ X z= aω uω ∧ (5) ω∈Qm,n
All GRs of V ∈ G(m, U) differ by a c ∈ F, c 6= 0 The coordinates of a deand are denoted by g(V). Vm composable vector z ∈ U, {aω , ω ∈ Qm,n } are known as the Pl¨ ucker coordinates (PC) of Vz . The lexicographically ordered set of PCs is completely determinedVby V to within c ∈ F. Note, that not m every z ∈ U, is necessarily decomposable; Vm if {aω , ω ∈ Qm,n } are the coordinates of z ∈ U, then z is decomposable if and only if the following conditions hold true (Marcus, 1973): m+1 X
(−1)k ai1 ,...,im−1 ,jk aj1 ,...,jk−1 ,jk+1 ,...,jm+1 = 0
k=1
(7) where 1 ≤ i1 < · · · < im−1 ≤ n and 1 ≤ j1 < j2 < · · · < jm+1 ≤ n. The set of quadratics defined by (7) are known as Quadratic Pl¨ ucker Relations (QPR) and describe an (n − m)m−dimensional algebraic variety, µ ¶ Ω(m, n), of the projective space m σ−1 P ,σ = , known as Grassmann variety. n Vm Vm The map ν : V ∈ G(m, U ) → V ∈ U expresses a natural injective correspondence between G(m, U) and 1-dimensional Vmsubspaces of Vm V the PCs U . By associating to every {aω , ω ∈ Qm,n }, the map ρ : G(m, U) → Pσ−1 is defined, and it is known as the Pl¨ ucker embedding (Griffiths and Harris, 1978) of G(m, U) in Pσ−1 ; the image of G(m, U) under ρ is Ω(m, n). For the rational vector space over R(s), XM , R(M (s)), a canonical polynomial (R[s]) GR may be defined and through that a basis free invariant of XM , the Pl¨ ucker matrix PM (Karcanias and Giannakopoulos, 1984); the rank properties of PM define the solvability conditions of the linear subproblem of R−DAP. An alternative test for decomposability that also allows a more convenient framework for computations is considered next. 4. THE GRASSMANN MATRIX AND DECOMPOSABILITY OF MULTIVECTORS Vm The Grassmann matrix of z ∈ U is introduced in this section and a number of its properties are examined. This matrix provides an alternative test for decomposability of z, which also allows the computation of the Vz solution space in an easy manner.
where {aω , ω ∈ Qm,n } are the coordinates of z Vm with respect to BUm . A vector z ∈ U is called ˜ such decomposable, if there exist v i ∈ U, i ∈ m that v1 ∧ . . . ∧ vm = z (6)
Lemma 1: (Marcus, 1973) Let U be an ndimensional vector space over F and let 0 6= z ∈ V m U . Then, z is decomposable, if and only if, there exists a set of linearly independent vectors ˜ in U such that {v i , i ∈ m}
The vector space Vz , spF {v i , i ∈ m} ˜ is called the It is known that if z, z 0 ∈ generating space of z. Vm U are nonzero and decomposable, then z 0 = c· z(c 6= 0) is equivalent to Vz = Vz0 ∈ G(m, U) and z is called a Grassmann Representative (GR) of Vz .
v i ∧ z = 0, ∀i ∈ m ˜
(8) ¦
This result is central in deriving the set of QPRs, as well as in deriving the alternative test.
Proposition 1: Let BU , {ui , i ∈ n ˜ } be a basis of U, BU = {u ∧, ω ∈ Q } the corresponding m,n ω Vm Pm basis of U and let v = t=1 ct ut , z = P ω∈Qm,n aω uω ∧ . Then, X
v∧z =
bγ uγ ∧, bγ =
γ∈Qm+1,n
m+1 X
(−1)k−1 cγ(k) aγ(k) ˆ
k=1
(9) where γ(k) denotes the k−th element of γ ∈ ˆ is the sequence (γ(1), . . . , γ(k − Qm+1,n and γ(k) 1), γ(k + 1), . . . , γ(m + 1)) ∈ Qm,n .
pair (m,Vn) and the coordinates {aω , ω ∈ Qm,n } m of z ∈ U and will be called the Grassmann matrix (GM) of z. Example 1: Let m = 2, n = 4 and {a12 , a13 , a , a23 , a24 , a34 } be the coordinates of z ∈ V142 U, dim U = 4, with respect to some basis. Then, a23 −a13 a12 0 ← (1, 2, 3) = γ1 a24 −a14 0 a12 ← (1, 2, 4) = γ2 2 Φ4 (z) = a34 0 −a14 a13 ← (1, 3, 4) = γ3 0 a34 −a24 a23 ← (2, 3, 4) = γ4
¦
Example 2: Let m = 2, n = 5 and {a12 , a13 , a14 , a15 , a23 , a24 , a25 , a34 , a35 , a45 } be the coordinates V2 of z ∈ U, dim U = 5, with respect to some basis. Then, a23 −a13 a12 0 0 ← (1, 2, 3) a24 −a14 0 a12 0 ← (1, 2, 4) a25 −a15 0 0 a12 ← (1, 2, 5) a34 0 −a14 a13 0 ← (1, 3, 4) a35 0 −a15 0 a13 ← (1, 3, 5) 2 ^ where k ∈ m + 1. Φ5 (z) = 0 −a15 a14 a45 0 ← (1, 4, 5) 0 a34 −a24 a23 0 ← (2, 3, 4) Definition 1: V Let {aω , ω ∈ Qm,n } be the coor m 0 a35 −a25 0 a23 ← (2, 3, 5) dinatesVof z ∈ U with respect to some basis m 0 a45 U , m + 1 ≤ n, γ = (j1 , . . . , jk , jm+1 ) ∈ BUm of 0 −a25 a24 ← (2, 4, 5) Qm+1,n and let ργ [ˆjk ] = (j1 , . . . , jk−1 , jk+1 , . . . , jm+1 ) ∈ 0 0 a45 −a35 a34 ← (3, 4, 5) Qγm,m+1 . We define the function φ : {i : i = Vm 1, . . . , n} × {γ, γ ∈ Qm+1,n } → F by: U The matrix Φm n (z) is defined for every z ∈ ( and the decomposability property of z is expressed φiγ , φγ (i) = 0, if i 6= γ by the following result. φiγ , φγ (i) = sign(jk : ργ [ˆjk ])aργ [ˆjk ] , if i = jk ∈ γ Theorem 1: Let U be an n−dimensional vecwhere sign(jk : ργ [ˆjk ]) = sign(jk , j1 , . . . , jk−1 , tor space over F, BU a basis of U , 0 6= z ∈ V m jk+1 , . . . , jm+1 ). U , Φm n (z) the GR of z with respect to BU and m let N (z) , Nr {Φm m n n (z)}, Then, Proposition 2: Let BU = {ui , i ∈ V n ˜ }, BU = m m , v = of U, {u (i) dim Nn (z) ≤ m and equality holds, if and Pnω ∧, ω ∈ Qm,n } be bases P c u ∈ U, v = 6 0, and z = a u ∧ ∈ only if z is decomposable. ω i i ω ω∈Qm,n Vmi=1 (ii) If dim Nnm (z) = m, then a representation of U, z 6= 0. v ∧ z = 0, if and only if the solution space, Vz , of v 1 ∧ . . . ∧ v m = z n X with respect to BU is given by Nnm (z). φiγ ci = 0, for all γ ∈ Qm+1,n (10) Notation: Let γ = (j1 , j2 , . . . , jk , jm+1 ) ∈ Qm+1,n , m + 1 ≤ n. We denote by Qγm,m+1 the subset of Qm,n sequences with elements taken from the γ set of integers. Qγm,m+1 has m + 1 elements and the sequences in it are defined from γ by deleting an index in γ. Thus, we may write: Qγm,m+1 , {ργ [ˆjk ] = (j1 , . . . , jk−1 , jk+1 , . . . , jm+1 )}
i=1
¦ If we denote by γt the elements of Qm+1,n (assumedµ to be ¶ lexicographically ordered), t = n 1, 2, . . . , = τ, then (10) may be exm+1 pressed in a matrix form as 1 c φγ1 , φ2γ1 , . . . , φiγ1 , . . . , φnγ1 1 .. .. .. .. c2 . . . . . 1 . φγ , φ2γ , . . . , φiγ , . . . , φnγ . = 0 (11) t t t c t i . .. .. .. .. . . . . .. φ1γ , φ2γτ , . . . , φiγτ , . . . , φnγτ {z } cn | τ | {z } m ,Φn (z)
Φm n (z)
,c τ ×n
The matrix ∈F is a structured matrix (has zeros in fixed positions), it is defined by the
¦ The above result provides an alternative characterisation for decomposability of multivectors,as well as a simple procedure for reconstruction of the solution space of the exterior equation. The matrix Φm n (z) that corresponds to a decomposable z will be referred to as canonical. Corollary 1: Let Φm n (z) be the GR of z ∈ V m U , z 6= 0. Then, (i) If m = 1, then for all n, Φ1n (z) is always canonical; furthermore, if n ≥ 3, rankF { Φ1n (z)} = n − 1. (ii) If m = n − 1, then Φn−1 (z) ∈ F 1×n and it is n always canonical with rankF { Φn−1 (z)} = 1. n (iii) If m = n − ρ, m > 1 and ρ ≥ 2, then for all z, rankF {Φm n (z)} ≥ n − m; equality holds, if and only if Φm n (z) is canonical.
¦ Note that parts (i), (ii) of the above result express the well known V1 result Vn−1 for decomposability of all vectors of U, U (Marcus, 1973). From part (iii) we also have: Vm Corollary 2: Let 0 6= z ∈ U, n − m ≥ m 2, m > 1. Φn (z) is canonical, if and only if Cn−m+1 {Φm n (z)} = 0. ¦ This result establishes the links between the new decomposability result based on Φm n (z) and the set of QPRs. The new decomposability test also provides an alternative characterisationµof ¶ the n Grassmann variety Ω(m, n) of Pσ−1 , σ = . m Vm Remark 2: Let z ∈ U, z 6= 0, and let P (z) be the point of Pσ−1 defined by the coordinates {aω , ω ∈ Qm,n } of z. P (z) ∈ Ω(m, n) if and only if the Grassmann matrix Φm n is canonical. ¦ In the next section we introduce the problem of “approximate decomposability” and give some partial results for the nontrivial case m = 2, n = 4.
a general point of Pρ from its Grassmann variety Ω(m, n). For the m = 2, n = 4, the Grassmannian G(2, 4) can be embedded via the Pl¨ ucker embedding in the projective space P(R6 ) = P5 as a four dimensional smooth projective variety defined by an QPR, i.e. a12 a34 − a13 a24 + a14 a23 = 0
(12)
where aij are the Pl¨ ucker coordinates of z ∈ P5 , where z = [a12 , a13 , a14 , a23 , a24 , a34 ]. We shall denote by p(z) = a12 a34 − a13 a24 + a14 a23 and by
Φ24 (z)
(13)
is the Grassmann matrix defined in (11)
a23 −a13 a12 0 a24 −a14 0 a12 Φ24 (z) = (14) a34 0 −a14 a13 0 a34 −a24 a23 Decomposability of z for this case implies that the rank of Φ24 (z) is 2 and thus the two smallest singular values σ3 , σ4 of Φ24 (z) must be both zero. We may calculate the singular values as follows: Proposition 3: The singular values of Φ24 are defined by the roots of the characteristic equation |λI − Φ24 (z)Φ24 (z)t | = {λ2 − ||z||2 λ + p(z)2 }2 (15)
5. THE GRASSMANN MATRIX AND APPROXIMATE DECOMPOSABILITY OF MULTIVECTORS The Grassmann matrix Φm n (z) has been defined as a structured matrix for any point of the projective µ ¶ n ρ − 1. It has been shown that space P , ρ = m its rank properties provide a characterisation of the Grassmann variety Ω(m, n) of Pρ and its null space define the solution of the exterior equation
and thus are of the form σ12 = σ22 = {||z||2 + {||z||4 − 4p(z)2 }1/2 } · 1/2 σ32 = σ42 = {||z||2 − {||z||4 − 4p(z)2 }1/2 } · 1/2 (16) ¦ Corollary 3: The two smallest singular values of Φ24 (z) are zero (σ3 = σ4 = 0), iff z is decomposable, that is p(z) = 0. In this σ1 = σ2 = ||z||. ¦
v 1 ∧ v 2 ∧ . . . ∧ v m = z, z ∈ ∧m U, v i ∈ U where U is an n-dimensional subspace over a field F. In fact, this matrix provides an equivalent characterisation to the QPR characterisation of Ω(m, n), which also leads to the reconstruction of the space V associated with the decomposable vector z. In this section we examine the special and nontrivial case m = 2, n = 4 and examine the properties of Φm n linked to the problem of “approximate decomposability” of multivectors defined in the following way: Problem Statement: Given z ∈ Pρ , which is not decomposable, define z˜ ∈ Pρ ∈ Ω(m, n) such that some distance d(z, z˜) is minimised. The above problem is referred to as the approximate decomposability problem (ADP) and it is essentially a problem of defining the distance of
The above result suggests that the smallest singular value of the Grassmann matrix plays a role in characterising the distance of z from the Grassmann variety Ω(m, n). The problem of approximate decomposability may be stated as follows: Given z ∈ P5 find k ∈ P5 such that k minimises Φ(z, k) = ||z − k||, subject to p(k) = 0
(17)
Note that if k min is the solution of the optimisation problem, then since ||z|| · sin θ = ||z − k min || 5
(18)
the direction k min ∈ P is the element of the embedded Grassmannian with the smallest angle from z ∈ P5 . The study of this optimisation problem can be done using the classical Langrangian method. The following result establishes the link of the third of the singular values of Φ24 (z) to the solution of the ADP.
Theorem 2: Given z ∈ P5 , the solution of the distance problem min ||z − k|| such that p(k) = 0 k
(19)
is given by the vector k ? , where k ? = µ3 (z)[I − σ32 (z)/p(z) · J]z
case is under investigation. The “approximate decomposability” notion allows the extension of the DAP framework to the area of studying approximate solutions of frequency assignment problems. This may provide the required tools for approximate pole assignment under constant feedback.
(20)
σ34 (z)/p(z)2 }
µ3 (z) = 1/{1 − (21) where σ3 (z) is the third singular value of Φ24 (z) and J is the matrix 0 0 00 0 1 0 0 0 0 −1 0 0 0 0 1 0 0 (22) J = 0 0 1 0 0 0 0 −1 0 0 0 0 1 0 00 0 0 ¦ The next result is central to the study of ADP and establishes the link between the optimal solution k ? and the evaluation of the distance of z from Ω(2, 4). Theorem 3: The distance of the point z ∈ P5 to the Grassmann variety Ω(2, 4) of P5 is exactly the third singular value σ3 of Φ24 (z). ¦ The results in this section are illustrated here: Example 3: Consider z = (10, 2, 15, 3, 1, −20)t ∈ P5 . This is not in Ω(2, 4) since p(z) = −157 6= 0, or σ3 = 5.917 6= 0 The decomposable k ? with the smallest distance from Ω(2, 4), equal to σ3 is the vector computed from (20) as k ? = [5.83, 1869, 16.489, 6.667, 0.582, −18.699]t 6. CONCLUSIONS A new criterion for decomposability of multivecVm tors of U has been given in terms of the rank properties of the Grassmann matrix Φm n (z) Vm defined for every z ∈ U. This test is simpler in nature to that given by the QPRs and it has the extra advantage that allows the reconstruction of the solution space Vz of the v 1 ∧ . . . ∧ v m = z equation, by computing the right null space of Φm n (z). The new formulation of decomposability in terms of the Grassmann matrix allows the study of the “approximate decomposability” as a distance problem in the projective space. Although the simple case of m = 2, n = 4 was considered here, this clearly demonstrates the significance of the new framework, since this distance is shown to be defined by a singular value of the Grassmann matrix. The extension of these results to the general
REFERENCES Brockett, R.W. and C.I. Byrnes (1981). Multivariable nyquist criterion, root loci and pole placement: A geometric viewpoint. IEEE Trans. Aut. Control 26, 271–283. Giannakopoulos, C., G. Kalogeropoulos and N. Karcanias (1983). The grassmann variety of nondynamic compensators and the determinantal assignment problem of linear systems. Bulletin of the Greek Mathematical Society 24, 35–57. Griffiths, P. and J. Harris (1978). Principles of Algebraic Geometry. John Wiley & Sons. New York. Hodge, W.V.D. and P.D. Pedoe (1952). Methods of Algebraic Geometry, Volume 2. Cambridge University Press. Karcanias, N. and C. Giannakopoulos (1984). Grassmann invariants, almost zeros and the determinantal pole-zero assignment problems of linear systems. Int. J. of Control 40, 673– 698. Karcanias, N. and C. Giannakopoulos (1989). Necessary and sufficient conditions for zero assignment by constant squaring down. Algebra and its Applications 122/123/124, 415– 446. Karcanias, N. and J. Leventides (2007). Grassmann matrices, decomposability and approximate solution of exterior equations. Control Engineering Centre Research Report. Leventides, J. and N. Karcanias (1995a). Global asymtotic linearisation of the pole placement map: A closed form solution of the output feedback problem. Automatica 31, 1303– 1309. Leventides, J. and N. Karcanias (1995b). Sufficient conditions for arbitrary pole assignment by constant decentralised output feedback. 8, 222–240. Marcus, M. (1973). Finite Dimensional Multilinear Algebra (Parts 1 and 2). Marcel Deker. New York. Marcus, M. and H. Minc (1964). A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon. Boston. Martin, C. and R. Hermann (1978). Applications of algebraic geometry to systems theory: The mcmillan degree and kronecker indices. SIAM Journal of Control and Optimisation 16, 743– 755.
Control Engineering Centre, School of Engineering and Mathematical Sciences, City University Northampton Square, London EC1V 0HB, UK [email protected] ∗∗ Department of Economics, Section of Mathematics and Informatics, University of Athens Pezmazoglou 8, Athens, Greece [email protected]
Vm Abstract: The exterior equation v 1 ∧ v 2 ∧ . . . ∧ v m = z, z ∈ U, v i ∈ U, U an n−dimensional vector space over F, is an integral part of the study of the Determinantal Assignment Problem (DAP) of linear systems and its solvability (decomposability of z) is characterised by the Quadratic Pl¨ ucker Relations (QPR). An alternative new test for decomposability of z is given, in terms of the rank properties of V the Grassmann matrix, Φm n (z), which is constructed by the m coordinates of z ∈ U. It is shown that the exterior equation is solvable (z is decomposable), if and only if dim Nnm (z) = m, where Nnm (z) , Nr {Φm n (z)}. If z ˜ is simply defined by is decomposable, then the solution space Vz = sp{v i , i ∈ m} Nnm (z). The linear algebra formulation of the decomposability problem provides an alternative framework (to that defined by the QPRs) for the study of solvability and computation of solutions of DAP and enables the definition and study of “approximate solutions” of exterior equations as a distance problem. For the case of m = 2, n = 4 a solution to approximate decomposability is given and its properties c are linked to the singular values of Φ24 (z). Copyright °2002 IFAC Keywords: Frequency Assignment Problems, Linear Algebra, Decomposability of Multivectors
1. INTRODUCTION The Determinantal Assignment Problem (DAP) has emerged as the abstract problem to which the study of pole, zero assignment of linear systems may be reduced (Karcanias and Giannakopoulos, 1984), (Leventides and Karcanias, 1995a). The multilinear nature of DAP suggests that the natural framework for its study is that of exterior algebra (Marcus, 1973). The study of DAP (Karcanias and Giannakopoulos, 1984) may be reduced to a
linear problem of zero assignment of polynomial combinants and a standard problem of multilinear algebra, that is the decomposability of multivectors (Marcus, 1973). The solution of the linear subproblem, whenever it exists, defines a linear space in a projective space P τ , whereas decomposability is characterised by the set of Quadratic Pl¨ ucker Relations (QPR), which define the Grassmann variety of P τ (Hodge and Pedoe, 1952), (Griffiths and Harris, 1978). Thus, solvability of
DAP is reduced to a problem of finding real intersections between the linear variety and the Grassmann variety of P τ . The importance of tools and techniques of algebraic geometry for control theory problems has been demonstrated by the work in (Martin and Hermann, 1978), (Brockett and Byrnes, 1981), (Leventides and Karcanias, 1995a), (Leventides and Karcanias, 1995b) etc. The approach adopted in (Karcanias and Giannakopoulos, 1984), (Karcanias and Giannakopoulos, 1989) differs from that in (Martin and Hermann, 1978), (Brockett and Byrnes, 1981) in the sense that the problem is studied in a projective, rather than an affine space setting; the former approach relies on exterior algebra and on the explicit description of the Grassmann variety, in terms of the QPRs, and has the advantage of being computational. The multilinear nature of DAP has been recently handled by a ”blow up” type methodology, using the notion of degenerate solution and known as ”Global Linearisation” (Leventides and Karcanias, 1995a). The aim of this paper is to give an alternative, linear algebra type, criterion for decomposability of multivectors to that defined by the QPRs and then introduce the notion of “approximate decomposability” that is an integral part of the effort of modifying suitably the overall framework to be able to handle issues of uncertainty. The problem Vm of decomposability of the multivector z ∈ U, where U is a vector space, is equivalent to the solvability of the exterior equation v 1 ∧ v 2 ∧ . . . ∧ v m = z, with v i ∈ U. The conditions for decomposability are given by the set of QPRs (Marcus, 1973) and the solution space Vz = sp{v i , i ∈ m} ˜ may be constructed as shown in (Giannakopoulos et al., 1983). The present approach handles simultaneously the question of decomposabilityV and m the reconstruction of Vz . For every z ∈ U with coordinates {aω , ω ∈ Qm,n }, the Grassmann matrix Φm n (z) of z is defined. It is shown, that rank{Φm (z)} ≥ n − m, for all z 6= 0, and z is n decomposable, if and only if, the equality sign holds. If rank{Φm n (z)} = n − m, then the solution space Vz is defined by Vz , Nr {Φm n (z)}. The rank based test for decomposability is easier to handle than the QPRs and provides a simple method for the computation of Vz . Furthermore, the matrix version formulation of the decomposability allows the definition of the notion of “approximate decomposability” that is studied for the one QPR case in the last section. Throughout the paper the following notation is adopted: If F is a field, or ring then F m×n denotes the set of m × n matrices over F. If H is a map, then R(H), Nr (H), Nl (H) denote the range, right, left nullspaces respectively. Qk,n denotes the set of lexicographically ordered, strictly increasing sequences of k integers from the set
n ˜ , {1, 2, . . . , n}. If V is a vector space and {v i1 , . . . , v ik } are vectors of V, then v i1 ∧ . . . ∧ v ik = v ω ∧, ω V = (i1 , . . . , ik ) denotes their exterior r product and V the r−th exterior power of V (Marcus, 1973). If H ∈ F m×n and r ≤ min{m, n}, then Cr (H) denotes the r−th compound matrix of H (Marcus, 1973). The proof of the results is given in (Karcanias and Leventides, 2007).
2. THE GENERAL DETERMINANTAL ASSIGNMENT PROBLEM Let M (s) ∈ Rm×l [s], m > l, rankR[s] {M (s)} = l and let H , {H(s) : H(s) ∈ Rl×m [s], rankR[s] {H(s)} = l}; the subset of H defined by all H ∈ Rl×m will be denoted by HR . Finding H ∈ H such that the polynomial fM (s, H) = det{H(s)M (s)}
(1)
has assigned zeros, is defined as the Determinantal Assignment Problem (DAP); if H ∈ HR , then the corresponding problem is defined as the constant DAP (R−DAP) (Karcanias and Giannakopoulos, 1984). The different versions of DAP have been introduced as the abstract unifying descriptions of frequency assignment problems (pole,zero) that arise in linear systems theory. The general case, DAP, covers the dynamic version of frequency assignment problems. If we require that fM (s, H) is an arbitrary Hurwitz polynomial, then different classes of Determinantal Stabilisation Problems (DSP) are defined. Let hti , mi , i ∈ ˜l be the rows of H ∈ H, columns of M (s). Then, (H) = ht1 ∧ . . . ∧ htl = ht ∧ ∈ µ Cl¶ m R1×q [s], q = , Cl (M (s)) = m1 (s) ∧ . . . ∧ l q ml (s) = m(s)∧ ∈ R [s] and by the Binet-Cauchy Theorem (Marcus and Minc, 1964) we have fM (s, H) = Cl (H)Cl (M (s)) =< h∧, m(s)∧ > and thus X fM (s, H) = hω mω (s) (2) ω∈Qm,n
where < ·, · > denotes scalar product, ω = (i1 , . . . , il ) ∈ Ql,m and hω , mω (s) are the entries in h∧, m(s)∧ respectively. Note that hω is the l ×l minor of H, which corresponds to the ω set of rows of H and thus is a multilinear alternating function of the hij entries of H. DAP may be reduced to a linear and a standard multilinear subproblem (Karcanias and Giannakopoulos, 1984): (i) Linear Subproblem of DAP: Let m(s)∧ = p(s) ∈ Rq [s]. Investigate the existence of k(s) ∈ Rq [s] such that for some given α(s) ∈ R[s], d = degα(s), and ed (s) = [1, s, . . . , sd ]t . fp (s, k) = k(s)t p(s) =
=
q X
ki (s)pi (s) = α(s) = αt ed (s)
(3)
i=1
(ii) Multilinear subproblem of DAP: Assume that for the given α(s) part (i) is solvable and let K(α) be the family of solutions. Determine whether there exists H ∈ H, H t = [h1 , . . . , hl ] such that h1 ∧ . . . ∧ hl = k, k ∈ K(α)
(4) ¦
fp (s, k), is defined for a given p(s), and it is called an R[s]−polynomial combinant, if ki ∈ R[s], and as R−polynomial combinant, if ki ∈ R. The solution of the exterior equation is a standard problem of exterior algebra, known as decomposability of multivectors (Marcus, 1973). The solvability of the linear subproblem is a standard problem of linear algebra. The solvability of (4) is characterised by the set of Quadratic Pl¨ ucker Relations (QPR) (Marcus, 1973), which in turn describe the Grassmann variety Ω(l, m) of Pq−1 (Hodge and Pedoe, 1952). Thus, solvability of R−DAP is equivalent to finding real intersections between Ψ(α) and Ω(l, m). The aim of this paper is to provide alternative criteria for solvability of (4), to those defined by the QPRs and introducing the notion of approximate solution of (4). A summary of key notions and results from exterior algebra are summarised first.
3. DECOMPOSABILITY OF MULTIVECTORS: BACKGROUND RESULTS Let U be a vector space over a field F and let G(m, U) be the Grassmannian (set of all m−dimensional subspaces of VpU). For Vp every VVp ∈ G(m, U) the injection map f : VV → U m V is a 1is well defined and if p = m, then Vm dimensional subspace U; if {v i , i ∈ m} ˜ is a Vm of is spanned by v 1 ∧. . .∧v m . Let basis of V, then BU = {ui , i ∈ n ˜ } be a basis of U and BUm = {uω ∧ : uω ∧ = ui1 ∧ . . . ∧ uim , ω = (i1 , . .V . , im ) ∈ Qm,n } m denote the corresponding basis of U. The genVm U may be expressed as eral vector z ∈ X z= aω uω ∧ (5) ω∈Qm,n
All GRs of V ∈ G(m, U) differ by a c ∈ F, c 6= 0 The coordinates of a deand are denoted by g(V). Vm composable vector z ∈ U, {aω , ω ∈ Qm,n } are known as the Pl¨ ucker coordinates (PC) of Vz . The lexicographically ordered set of PCs is completely determinedVby V to within c ∈ F. Note, that not m every z ∈ U, is necessarily decomposable; Vm if {aω , ω ∈ Qm,n } are the coordinates of z ∈ U, then z is decomposable if and only if the following conditions hold true (Marcus, 1973): m+1 X
(−1)k ai1 ,...,im−1 ,jk aj1 ,...,jk−1 ,jk+1 ,...,jm+1 = 0
k=1
(7) where 1 ≤ i1 < · · · < im−1 ≤ n and 1 ≤ j1 < j2 < · · · < jm+1 ≤ n. The set of quadratics defined by (7) are known as Quadratic Pl¨ ucker Relations (QPR) and describe an (n − m)m−dimensional algebraic variety, µ ¶ Ω(m, n), of the projective space m σ−1 P ,σ = , known as Grassmann variety. n Vm Vm The map ν : V ∈ G(m, U ) → V ∈ U expresses a natural injective correspondence between G(m, U) and 1-dimensional Vmsubspaces of Vm V the PCs U . By associating to every {aω , ω ∈ Qm,n }, the map ρ : G(m, U) → Pσ−1 is defined, and it is known as the Pl¨ ucker embedding (Griffiths and Harris, 1978) of G(m, U) in Pσ−1 ; the image of G(m, U) under ρ is Ω(m, n). For the rational vector space over R(s), XM , R(M (s)), a canonical polynomial (R[s]) GR may be defined and through that a basis free invariant of XM , the Pl¨ ucker matrix PM (Karcanias and Giannakopoulos, 1984); the rank properties of PM define the solvability conditions of the linear subproblem of R−DAP. An alternative test for decomposability that also allows a more convenient framework for computations is considered next. 4. THE GRASSMANN MATRIX AND DECOMPOSABILITY OF MULTIVECTORS Vm The Grassmann matrix of z ∈ U is introduced in this section and a number of its properties are examined. This matrix provides an alternative test for decomposability of z, which also allows the computation of the Vz solution space in an easy manner.
where {aω , ω ∈ Qm,n } are the coordinates of z Vm with respect to BUm . A vector z ∈ U is called ˜ such decomposable, if there exist v i ∈ U, i ∈ m that v1 ∧ . . . ∧ vm = z (6)
Lemma 1: (Marcus, 1973) Let U be an ndimensional vector space over F and let 0 6= z ∈ V m U . Then, z is decomposable, if and only if, there exists a set of linearly independent vectors ˜ in U such that {v i , i ∈ m}
The vector space Vz , spF {v i , i ∈ m} ˜ is called the It is known that if z, z 0 ∈ generating space of z. Vm U are nonzero and decomposable, then z 0 = c· z(c 6= 0) is equivalent to Vz = Vz0 ∈ G(m, U) and z is called a Grassmann Representative (GR) of Vz .
v i ∧ z = 0, ∀i ∈ m ˜
(8) ¦
This result is central in deriving the set of QPRs, as well as in deriving the alternative test.
Proposition 1: Let BU , {ui , i ∈ n ˜ } be a basis of U, BU = {u ∧, ω ∈ Q } the corresponding m,n ω Vm Pm basis of U and let v = t=1 ct ut , z = P ω∈Qm,n aω uω ∧ . Then, X
v∧z =
bγ uγ ∧, bγ =
γ∈Qm+1,n
m+1 X
(−1)k−1 cγ(k) aγ(k) ˆ
k=1
(9) where γ(k) denotes the k−th element of γ ∈ ˆ is the sequence (γ(1), . . . , γ(k − Qm+1,n and γ(k) 1), γ(k + 1), . . . , γ(m + 1)) ∈ Qm,n .
pair (m,Vn) and the coordinates {aω , ω ∈ Qm,n } m of z ∈ U and will be called the Grassmann matrix (GM) of z. Example 1: Let m = 2, n = 4 and {a12 , a13 , a , a23 , a24 , a34 } be the coordinates of z ∈ V142 U, dim U = 4, with respect to some basis. Then, a23 −a13 a12 0 ← (1, 2, 3) = γ1 a24 −a14 0 a12 ← (1, 2, 4) = γ2 2 Φ4 (z) = a34 0 −a14 a13 ← (1, 3, 4) = γ3 0 a34 −a24 a23 ← (2, 3, 4) = γ4
¦
Example 2: Let m = 2, n = 5 and {a12 , a13 , a14 , a15 , a23 , a24 , a25 , a34 , a35 , a45 } be the coordinates V2 of z ∈ U, dim U = 5, with respect to some basis. Then, a23 −a13 a12 0 0 ← (1, 2, 3) a24 −a14 0 a12 0 ← (1, 2, 4) a25 −a15 0 0 a12 ← (1, 2, 5) a34 0 −a14 a13 0 ← (1, 3, 4) a35 0 −a15 0 a13 ← (1, 3, 5) 2 ^ where k ∈ m + 1. Φ5 (z) = 0 −a15 a14 a45 0 ← (1, 4, 5) 0 a34 −a24 a23 0 ← (2, 3, 4) Definition 1: V Let {aω , ω ∈ Qm,n } be the coor m 0 a35 −a25 0 a23 ← (2, 3, 5) dinatesVof z ∈ U with respect to some basis m 0 a45 U , m + 1 ≤ n, γ = (j1 , . . . , jk , jm+1 ) ∈ BUm of 0 −a25 a24 ← (2, 4, 5) Qm+1,n and let ργ [ˆjk ] = (j1 , . . . , jk−1 , jk+1 , . . . , jm+1 ) ∈ 0 0 a45 −a35 a34 ← (3, 4, 5) Qγm,m+1 . We define the function φ : {i : i = Vm 1, . . . , n} × {γ, γ ∈ Qm+1,n } → F by: U The matrix Φm n (z) is defined for every z ∈ ( and the decomposability property of z is expressed φiγ , φγ (i) = 0, if i 6= γ by the following result. φiγ , φγ (i) = sign(jk : ργ [ˆjk ])aργ [ˆjk ] , if i = jk ∈ γ Theorem 1: Let U be an n−dimensional vecwhere sign(jk : ργ [ˆjk ]) = sign(jk , j1 , . . . , jk−1 , tor space over F, BU a basis of U , 0 6= z ∈ V m jk+1 , . . . , jm+1 ). U , Φm n (z) the GR of z with respect to BU and m let N (z) , Nr {Φm m n n (z)}, Then, Proposition 2: Let BU = {ui , i ∈ V n ˜ }, BU = m m , v = of U, {u (i) dim Nn (z) ≤ m and equality holds, if and Pnω ∧, ω ∈ Qm,n } be bases P c u ∈ U, v = 6 0, and z = a u ∧ ∈ only if z is decomposable. ω i i ω ω∈Qm,n Vmi=1 (ii) If dim Nnm (z) = m, then a representation of U, z 6= 0. v ∧ z = 0, if and only if the solution space, Vz , of v 1 ∧ . . . ∧ v m = z n X with respect to BU is given by Nnm (z). φiγ ci = 0, for all γ ∈ Qm+1,n (10) Notation: Let γ = (j1 , j2 , . . . , jk , jm+1 ) ∈ Qm+1,n , m + 1 ≤ n. We denote by Qγm,m+1 the subset of Qm,n sequences with elements taken from the γ set of integers. Qγm,m+1 has m + 1 elements and the sequences in it are defined from γ by deleting an index in γ. Thus, we may write: Qγm,m+1 , {ργ [ˆjk ] = (j1 , . . . , jk−1 , jk+1 , . . . , jm+1 )}
i=1
¦ If we denote by γt the elements of Qm+1,n (assumedµ to be ¶ lexicographically ordered), t = n 1, 2, . . . , = τ, then (10) may be exm+1 pressed in a matrix form as 1 c φγ1 , φ2γ1 , . . . , φiγ1 , . . . , φnγ1 1 .. .. .. .. c2 . . . . . 1 . φγ , φ2γ , . . . , φiγ , . . . , φnγ . = 0 (11) t t t c t i . .. .. .. .. . . . . .. φ1γ , φ2γτ , . . . , φiγτ , . . . , φnγτ {z } cn | τ | {z } m ,Φn (z)
Φm n (z)
,c τ ×n
The matrix ∈F is a structured matrix (has zeros in fixed positions), it is defined by the
¦ The above result provides an alternative characterisation for decomposability of multivectors,as well as a simple procedure for reconstruction of the solution space of the exterior equation. The matrix Φm n (z) that corresponds to a decomposable z will be referred to as canonical. Corollary 1: Let Φm n (z) be the GR of z ∈ V m U , z 6= 0. Then, (i) If m = 1, then for all n, Φ1n (z) is always canonical; furthermore, if n ≥ 3, rankF { Φ1n (z)} = n − 1. (ii) If m = n − 1, then Φn−1 (z) ∈ F 1×n and it is n always canonical with rankF { Φn−1 (z)} = 1. n (iii) If m = n − ρ, m > 1 and ρ ≥ 2, then for all z, rankF {Φm n (z)} ≥ n − m; equality holds, if and only if Φm n (z) is canonical.
¦ Note that parts (i), (ii) of the above result express the well known V1 result Vn−1 for decomposability of all vectors of U, U (Marcus, 1973). From part (iii) we also have: Vm Corollary 2: Let 0 6= z ∈ U, n − m ≥ m 2, m > 1. Φn (z) is canonical, if and only if Cn−m+1 {Φm n (z)} = 0. ¦ This result establishes the links between the new decomposability result based on Φm n (z) and the set of QPRs. The new decomposability test also provides an alternative characterisationµof ¶ the n Grassmann variety Ω(m, n) of Pσ−1 , σ = . m Vm Remark 2: Let z ∈ U, z 6= 0, and let P (z) be the point of Pσ−1 defined by the coordinates {aω , ω ∈ Qm,n } of z. P (z) ∈ Ω(m, n) if and only if the Grassmann matrix Φm n is canonical. ¦ In the next section we introduce the problem of “approximate decomposability” and give some partial results for the nontrivial case m = 2, n = 4.
a general point of Pρ from its Grassmann variety Ω(m, n). For the m = 2, n = 4, the Grassmannian G(2, 4) can be embedded via the Pl¨ ucker embedding in the projective space P(R6 ) = P5 as a four dimensional smooth projective variety defined by an QPR, i.e. a12 a34 − a13 a24 + a14 a23 = 0
(12)
where aij are the Pl¨ ucker coordinates of z ∈ P5 , where z = [a12 , a13 , a14 , a23 , a24 , a34 ]. We shall denote by p(z) = a12 a34 − a13 a24 + a14 a23 and by
Φ24 (z)
(13)
is the Grassmann matrix defined in (11)
a23 −a13 a12 0 a24 −a14 0 a12 Φ24 (z) = (14) a34 0 −a14 a13 0 a34 −a24 a23 Decomposability of z for this case implies that the rank of Φ24 (z) is 2 and thus the two smallest singular values σ3 , σ4 of Φ24 (z) must be both zero. We may calculate the singular values as follows: Proposition 3: The singular values of Φ24 are defined by the roots of the characteristic equation |λI − Φ24 (z)Φ24 (z)t | = {λ2 − ||z||2 λ + p(z)2 }2 (15)
5. THE GRASSMANN MATRIX AND APPROXIMATE DECOMPOSABILITY OF MULTIVECTORS The Grassmann matrix Φm n (z) has been defined as a structured matrix for any point of the projective µ ¶ n ρ − 1. It has been shown that space P , ρ = m its rank properties provide a characterisation of the Grassmann variety Ω(m, n) of Pρ and its null space define the solution of the exterior equation
and thus are of the form σ12 = σ22 = {||z||2 + {||z||4 − 4p(z)2 }1/2 } · 1/2 σ32 = σ42 = {||z||2 − {||z||4 − 4p(z)2 }1/2 } · 1/2 (16) ¦ Corollary 3: The two smallest singular values of Φ24 (z) are zero (σ3 = σ4 = 0), iff z is decomposable, that is p(z) = 0. In this σ1 = σ2 = ||z||. ¦
v 1 ∧ v 2 ∧ . . . ∧ v m = z, z ∈ ∧m U, v i ∈ U where U is an n-dimensional subspace over a field F. In fact, this matrix provides an equivalent characterisation to the QPR characterisation of Ω(m, n), which also leads to the reconstruction of the space V associated with the decomposable vector z. In this section we examine the special and nontrivial case m = 2, n = 4 and examine the properties of Φm n linked to the problem of “approximate decomposability” of multivectors defined in the following way: Problem Statement: Given z ∈ Pρ , which is not decomposable, define z˜ ∈ Pρ ∈ Ω(m, n) such that some distance d(z, z˜) is minimised. The above problem is referred to as the approximate decomposability problem (ADP) and it is essentially a problem of defining the distance of
The above result suggests that the smallest singular value of the Grassmann matrix plays a role in characterising the distance of z from the Grassmann variety Ω(m, n). The problem of approximate decomposability may be stated as follows: Given z ∈ P5 find k ∈ P5 such that k minimises Φ(z, k) = ||z − k||, subject to p(k) = 0
(17)
Note that if k min is the solution of the optimisation problem, then since ||z|| · sin θ = ||z − k min || 5
(18)
the direction k min ∈ P is the element of the embedded Grassmannian with the smallest angle from z ∈ P5 . The study of this optimisation problem can be done using the classical Langrangian method. The following result establishes the link of the third of the singular values of Φ24 (z) to the solution of the ADP.
Theorem 2: Given z ∈ P5 , the solution of the distance problem min ||z − k|| such that p(k) = 0 k
(19)
is given by the vector k ? , where k ? = µ3 (z)[I − σ32 (z)/p(z) · J]z
case is under investigation. The “approximate decomposability” notion allows the extension of the DAP framework to the area of studying approximate solutions of frequency assignment problems. This may provide the required tools for approximate pole assignment under constant feedback.
(20)
σ34 (z)/p(z)2 }
µ3 (z) = 1/{1 − (21) where σ3 (z) is the third singular value of Φ24 (z) and J is the matrix 0 0 00 0 1 0 0 0 0 −1 0 0 0 0 1 0 0 (22) J = 0 0 1 0 0 0 0 −1 0 0 0 0 1 0 00 0 0 ¦ The next result is central to the study of ADP and establishes the link between the optimal solution k ? and the evaluation of the distance of z from Ω(2, 4). Theorem 3: The distance of the point z ∈ P5 to the Grassmann variety Ω(2, 4) of P5 is exactly the third singular value σ3 of Φ24 (z). ¦ The results in this section are illustrated here: Example 3: Consider z = (10, 2, 15, 3, 1, −20)t ∈ P5 . This is not in Ω(2, 4) since p(z) = −157 6= 0, or σ3 = 5.917 6= 0 The decomposable k ? with the smallest distance from Ω(2, 4), equal to σ3 is the vector computed from (20) as k ? = [5.83, 1869, 16.489, 6.667, 0.582, −18.699]t 6. CONCLUSIONS A new criterion for decomposability of multivecVm tors of U has been given in terms of the rank properties of the Grassmann matrix Φm n (z) Vm defined for every z ∈ U. This test is simpler in nature to that given by the QPRs and it has the extra advantage that allows the reconstruction of the solution space Vz of the v 1 ∧ . . . ∧ v m = z equation, by computing the right null space of Φm n (z). The new formulation of decomposability in terms of the Grassmann matrix allows the study of the “approximate decomposability” as a distance problem in the projective space. Although the simple case of m = 2, n = 4 was considered here, this clearly demonstrates the significance of the new framework, since this distance is shown to be defined by a singular value of the Grassmann matrix. The extension of these results to the general
REFERENCES Brockett, R.W. and C.I. Byrnes (1981). Multivariable nyquist criterion, root loci and pole placement: A geometric viewpoint. IEEE Trans. Aut. Control 26, 271–283. Giannakopoulos, C., G. Kalogeropoulos and N. Karcanias (1983). The grassmann variety of nondynamic compensators and the determinantal assignment problem of linear systems. Bulletin of the Greek Mathematical Society 24, 35–57. Griffiths, P. and J. Harris (1978). Principles of Algebraic Geometry. John Wiley & Sons. New York. Hodge, W.V.D. and P.D. Pedoe (1952). Methods of Algebraic Geometry, Volume 2. Cambridge University Press. Karcanias, N. and C. Giannakopoulos (1984). Grassmann invariants, almost zeros and the determinantal pole-zero assignment problems of linear systems. Int. J. of Control 40, 673– 698. Karcanias, N. and C. Giannakopoulos (1989). Necessary and sufficient conditions for zero assignment by constant squaring down. Algebra and its Applications 122/123/124, 415– 446. Karcanias, N. and J. Leventides (2007). Grassmann matrices, decomposability and approximate solution of exterior equations. Control Engineering Centre Research Report. Leventides, J. and N. Karcanias (1995a). Global asymtotic linearisation of the pole placement map: A closed form solution of the output feedback problem. Automatica 31, 1303– 1309. Leventides, J. and N. Karcanias (1995b). Sufficient conditions for arbitrary pole assignment by constant decentralised output feedback. 8, 222–240. Marcus, M. (1973). Finite Dimensional Multilinear Algebra (Parts 1 and 2). Marcel Deker. New York. Marcus, M. and H. Minc (1964). A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon. Boston. Martin, C. and R. Hermann (1978). Applications of algebraic geometry to systems theory: The mcmillan degree and kronecker indices. SIAM Journal of Control and Optimisation 16, 743– 755.