Generic degree structure of the minimal polynomial nullspace basis: a block Toeplitz matrix approach

 Jack Norton
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1 Generic degree structure of the minimal polynomial nullspace basis: a block Toeplitz matrix approach Bhaskar Ramasubramanian 1, Swanand R. Khare and Madhu N. Belur 3 December 17, Department of Electrical and Computer Engineering, University of Maryland, College Park, MD, USA. Department of Mathematics, Indian Institute of Technology, Kharagpur, India. 3 Department of Electrical Engineering, Indian Institute of Technology Bombay, India. 1 / 13
2 Motivation / 13 Goal : Computing the minimal polynomial basis (MPB) of a given polynomial matrix. Current state of the art: Involves explicit knowledge of entries of polynomial matrix. Examples  matrix pencils, LQ factorization of Toeplitz matrices. Numerically robust algorithms the key. This work: Generic case  only work with degrees of entries in given polynomial matrix to determine degrees of entries in MPB. For specific case, get upper bound on degree structure of MPB. No numerical considerations since problem transformed to computing annihilator of a constant matrix.
3 Preliminaries  Minimal Polynomial Basis 3 / 13 R[s]  polynomials in s with real coefficients R m n [s]  m n matrix with entries from R[s]. (m < n) d  degree of polynomial vector, i.e., maximum among degrees of polynomial components. {q(s) R n [s] R(s)q(s) = 0}  nullspace of R(s) R m n [s]. Minimal Polynomial Basis Given R R m n [s], of degree d and rank m, let q 1, q,..., q n m be polynomially independent vectors in the nullspace of R arranged in nondecreasing order of their degrees a 1, a,..., a n m respectively. Then, the set {q 1, q,..., q n m } is an MPB if for any other set of n m polynomially independent vectors from the nullspace with degrees b 1 b b n m, it turns out that a i b i for i = 1,,..., n m. MPB is unique upto degrees of the polynomial vectors.
4 Preliminaries  Genericity of Parameters 4 / 13 Algebraic variety  set of solutions, E q R n to a system of polynomial equations. Zero equation in variables renders variety trivial. Nontrivial algebraic variety in R n is a set of measure zero. Genericity Property P in terms of variables p 1, p,..., p n R is said to be satisfied generically if the set of values p 1, p,..., p n that do not satisfy P form a nontrivial algebraic variety in R n. Examples Two nonzero polynomials are generically coprime. Entries of a square matrix chosen generically from R makes it nonsingular. Left primeness of a wide polynomial matrix.
5 Problem Formulation 5 / 13 Given R R m n [s], define D Z m n such that [D] ij := deg[r] ij. If [R] ij = 0, then [D] ij :=. Define the sets: Z + = {z Z z 0} Z + = Z + { } Can construct unique D Z m n + for given R R m n [s] This D is called the degree structure of R. Given D, there exist many R with the same degree structure. D(R) := {R R m n [s] with degree structure D Z m n + }
6 Problem Formulation, Main Result 6 / 13 Problem 1 Given R R m n [s] with degree structure D Z m n +, let Q R n (n m) [s] be such that the columns of Q form an MPB for the nullspace of R. Let K Z n (n m) + denote the degree structure of Q. Then, can we determine K from D? Problem Let R 1, R D(R) and Q 1, Q R n (n m) [s] be such that the columns of Q 1 and Q form MPBs for R 1 and R respectively. Let K 1 and K denote the degree structures of Q 1 and Q respectively. Then, is K 1 = K generically? Theorem 1 Given degree structures K 1, K of minimal polynomial bases corresponding to the same degree structure D, K 1 = K.
7 Toeplitz Matrices 7 / 13 Given R R m n [s] with degree d, R = R 0 + R 1 s + + R d s d where R i R m n for i = 0, 1,..., d. Construct a sequence of real structured matrices from the given polynomial matrix as: R 0 A R 1 A A 0 =.,A 1 =, A = 0 0 A 0 A 0, (1) 0 R d 0 0 A 0 where 0 s in the above equation are zero matrices of size m n and A i R (d+i+1)m (i+1)n. Stop when (d + i + 1)m (i + 1)n. (Right) kernels of A i related to nullspace of R.
8 Degree Structure of MPB of 1 3 Polynomial Matrix Let R R 1 3 [s] have degree structure D = [ a b c ] and let columns of Q R 3 [s] with deg struct K form an MPB of R. Assume: a b c; 0 / R; b c. Theorem For even c, the degree structure of the MPB is given by: c c K = c c () b c b c The degree structure of the MPB for odd c is given by K = c 1 c 1 c 1 (c b) c+1 c+1 c+1 (c b) (3) When c = b + k, the MPB will contain the zero polynomial, corresponding to a term in its degree structure. 8 / 13
9 Example 9 / 13 Given D = [ 0 1 ], find K = A 0 = ; A 1 =, such that DK = A 1 is a wide matrix. We have (D 0 + D 1 s + D s )K = 0, where D i corresponds to the coefficients of the degree i terms in the given polynomial matrix. Note: If a particular A i yields only some columns of the MPB, the remaining columns can be got by constructing A i+1.
10 Example 10 / 13 Need: Constant matrix P such that A 1 P = 0. For last row of A 1 to be annihilated, corresponding element(s) in K must be zero. This effectively eliminates the last column of A 1, as shown below: P = 0 0, yielding K =
11 Saturation 11 / 13 Observe: degree structure of K independent of a. Same algorithm can be used to determine degree structure of minimal left indices of K, D 1. If D D 1 (component wise), and they have an MPB with the same degree structure, K, then D is said to be saturated. Unsaturated D some degree of freedom to change one or more coefficients from 0 to a nonzero value,. Saturation: degree of freedom offered to replace zeros by nonzeros in degree structure of D while maintaining that of K. Proposition When D = [ a b c ] Z 1 3 +, and c b, D sat = [ b b c ]. D sat for higher dimensions of D? Not (yet) known.
12 Conclusions 1 / 13 Degree structure of MPB of a given polynomial matrix depends only on its degree structure, and not its coefficients. Exact degree structure of MPB got for generic case; same serves as upper bound for specific case. MPB of 1 3 degree structure studied in detail. Closed form solution for degree structure of MPB provided. Genericity of parameters ensured matrices had full rank. Saturation of a degree structure examined in the context of freedom of making some zero coefficients nonzero. Complete absence of extensive numerical computations  results depended only on zero/ nonzero nature of coefficients.
13 Thank You 13 / 13