Almost Strictly Sign Regular matrices 1 Introduction

Proceedings of the 13th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2013 24–27 June, 2013.

Almost Strictly Sign Regular matrices Pedro Alonso1 , Juan Manuel Pe˜ na2 and Mar´ıa Luisa Serrano1 1 2

Departamento de Matem´ aticas, Universidad de Oviedo, Spain

Departamento de Matem´ atica Aplicada, Universidad de Zaragoza, Spain

emails: [email protected], [email protected], [email protected]

Abstract In this work we announce a characterization of almost strictly sign regular matrices using Neville elimination. Neville elimination is an elimination procedure that consists of making zeros in a column of a matrix by adding to each row an adequate multiple of the previous one. Key words: Almost Strictly Sign Regular matrices, Neville elimination, characterization MSC 2000: AMS codes (optional)

1

Introduction

Totally positive (TP) and sign regular (SR) matrices arise naturally in many areas of mathematics, statistics, economics, etc (see [3], [4], [10] and [11]). On the other hand, in some recent papers it has been shown that an elimination procedure, called Neville elimination, is very convenient when working with totally positive and sign regular matrices. Many properties of these matrices and its subclasses have been proved using this elimination procedure (see [6] and [7]). Roughly speaking, Neville elimination consists of making zeros in a column of a matrix by adding to each row an adequate multiple of the previous one (see [5] for more details), instead of using just a row with a fixed pivot as in Gauss elimination. This process is an alternative to Gaussian elimination which has been proved to be very useful with totally positive matrices, sign regular matrices or other related types of matrices, without increasing the error bounds (see [2]). Furthermore, we show that using this procedure it is possible to obtain algorithms with high relative accuracy (HRA) for the computation of eigenvalues

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Almost Strictly Sign Regular matrices

and inverses of Pascal matrices (total nonnegative matrices) (see [1]). Other related results on HRA can be seen in [9]. In the last years different authors have studied the matrices strictly totally positive (STP) matrices, almost strictly totally positive (ASTP) matrices (see [7]), strictly sign regular (SSR) matrices or almost strictly sign regular (ASSR) matrices (see [8]). In this work we announce some results about ASSR matrices. The main result characterizes ASSR matrices using Neville elimination.

2

Basic notations and auxiliary results

For k, n ∈ N, with 1 ≤ k ≤ n, Qk,n denotes the set of all increasing sequences of k natural numbers not greater than n. For α = (α1 , . . . , αk ), β = (β1 , . . . , βk ) ∈ Qk,n and A an n × n real matrix, we denote A[α|β] the k × k submatrix of A containing rows α1 , . . . , αk and columns β1 , . . . , βk of A. If α = β, we denote by A[α] := A[α|α] the corresponding principal minor. Q0k,n denotes the set of increasing sequences of k consecutive natural numbers not greater than n. First we define the matrices type-I and type-II staircase. Definiton 1 A matrix A = (aij )1≤i,j≤n is called type-I staircase if it satisfies simultaneously the following conditions • a11 6= 0, a22 6= 0, . . . , ann 6= 0; • aij = 0, i > j ⇒ akl = 0, ∀l ≤ j, i ≤ k; • aij = 0, i < j ⇒ akl = 0, ∀k ≤ i, j ≤ l. Definiton 2 A matrix A = (aij )1≤i,j≤n is called type-II staircase if it satisfies simultaneously the following conditions • a1,n 6= 0, a2,n−1 6= 0, . . . , an,1 6= 0; • aij = 0, j > n − i + 1 ⇒ akl = 0, ∀i ≤ k, j ≤ l; • aij = 0, j < n − i + 1 ⇒ akl = 0, ∀k ≤ i, l ≤ j. To describe clearly the zero pattern of a nonsingular matrix A type-I staircase (or type-II staircase, using the n × n backward identity matrix Pn ) we must introduce some notations. For a matrix A = (aij )1≤i,j≤n type-I staircase, we denote i0 = 1,

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j0 = 1,

(1)

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˜ a, M.L. Serrano P. Alonso, J. M. Pen

for k = 1, . . . , l:  ik = max i / aijk−1 6= 0 + 1 (≤ n + 1),

(2)

jk = max {j / aik j = 0} + 1 (≤ n + 1),

(3)

where l is given in this recurrent definition by jl = n + 1. Analogously we denote b bi0 = 1 j0 = 1,

(4)

for k = 1, . . . , r: n o b jk = max j / abik−1 j 6= 0 + 1 (≤ n + 1), n o bik = max i / a b = 0 + 1 (≤ n + 1), ijk

(5) (6)

where bir = n + 1. Finally, we denote by I, J, Ib and Jb the following sets of indices, thereby defining the zero pattern in the matrix A: I = n{i0 , i1 , . . . , il }o, Ib = bi0 , bi1 , . . . , bir ,

J = n{j0 , j1 , . . . , jl }o, b Jb = j0 , b j1 , . . . , b jr .

Definiton 3 Given a matrix A = (aij )1≤i,j≤n type-I (type-II) staircase, we say that a submatrix A[α|β], with α, β ∈ Qm,n is nontrivial if all its main diagonal (secondary diagonal) elements are non-zero. The minor associated to a nontrivial submatrix (A[α|β]) is called nontrivial minor (det A[α|β]). Next, we define the SR, SSR and ASSR matrices, and finally we present the characterization performed in [8] for ASSR matrices (see Theorem 10, pp 4184). Definiton 4 Given a vector ε = (ε1 , ε2 , . . . , εn ) ∈ Rn , we say that ε is a signature sequence, or simply, is a signature, if |εi | = | ± 1| = 1, ∀i ∈ N, i ≤ n. Definiton 5 A real matrix A n × n is said to be SR, with signature ε = (ε1 , ε2 , . . . , εn ), if all its minors satisfy that εm det A[α|β] ≥ 0,

α, β ∈ Qm,n ,

m ≤ n.

(7)

Definiton 6 A real matrix A n × n is said to be SSR, with signature ε = (ε1 , ε2 , . . . , εn ), if all its minors satisfy that εm det A[α|β] > 0,

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α, β ∈ Qm,n ,

m ≤ n.

(8)

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Almost Strictly Sign Regular matrices

Definiton 7 A real matrix A n × n is said to be ASSR, with signature ε = (ε1 , ε2 , . . . , εn ), if all its nontrivial minors det A[α|β] satisfy that εm det A[α|β] > 0,

α, β ∈ Qm,n ,

m ≤ n.

(9)

Theorem 1 Let A a real matrix n × n and ε = (ε1 , ε2 , . . . , εn ) be a signature. Then A is nonsingular ASSR with signature ε if and only if A is a type-I or type-II staircase matrix, and all its nontrivial minors with α, β ∈ Q0m,n , m ≤ n, satisfy εm det A[α|β] > 0.

3

(10)

Main results

Let A = (aij )1≤i,j≤n be a n × n matrix and h = 1, . . . n − 1, we denote by Ah the matrix defined as Ah := (ahij )1≤i,j≤n−h+1 , ahij := ai+h−1,j+h−1 . (11) Analogously, the transpose of Ah , i.e. ATh , is denoted as ATh := (aT,h ij )1≤i,j≤n−h+1 ,

aT,h ij := aj+h−1,i+h−1 .

(12)

In the following two results, we announce necessary conditions for nonsingular ASSR type-I and type-II staircase matrices, respectively. Theorem 2 Let B = (bij )1≤i,j≤n be a nonsingular type-I staircase matrix, with zero pattern b J. b If B is ASSR with signature ε = (ε1 , ε2 , . . . , εn ), then the Neville defined by I, J, I, elimination of B can be performed without row exchanges and the pivots pij satisfy, for any 1 ≤ j ≤ i ≤ n, pij = 0 ⇔ bij = 0 (13) εj−jt εj−jt +1 pij > 0 ⇔ bij 6= 0

(14)

jt = max {js / 0 ≤ s ≤ k − 1, j − js ≤ i − is }

(15)

where ε0 = 1, and k is the only index satisfying that jk−1 ≤ j < jk . Theorem 3 Let B = (bij )1≤i,j≤n be a nonsingular type-II staircase matrix, with zero patb J. b If B is ASSR with signature ε = (ε1 , ε2 , . . . , εn ), then the Neville tern defined by I, J, I, T elimination of B can be performed without row exchanges and the pivots qij satisfy, for any 1 ≤ i ≤ j ≤ n, qij = 0 ⇔ bij = 0 (16)

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˜ a, M.L. Serrano P. Alonso, J. M. Pen

where ε0 = 1,

εi−bit εi−bit +1 qij > 0 ⇔ bij 6= 0

(17)

n o bit = max bis / 0 ≤ s ≤ k − 1, i − bis ≤ j − b js

(18)

and k is the only index satisfying that bik−1 ≤ i < bik . Finally, in the following two results we characterize all ASSR matrices through Neville elimination. Theorem 4 A nonsingular matrix A = (aij )1≤i,j≤n is ASSR with signature ε = (ε1 , ε2 , . . . , εn ), with ε2 = 1 if and only if for every h = 1, . . . , n − 1 the following properties hold simultaneously: (i) A is type-I staircase; (ii) the Neville elimination of the matrices Ah and ATh can be performed without row exchanges; (iii) the pivots phij of the Neville elimination of Ah satisfy conditions corresponding to (13), h of the Neville elimination AT satisfy (16) and (17); (14), and the pivots qij h (iv) for the positions (ih , j h ) of matrix Ah : • if ih ≥ j h and ih − j h = iht − jth then εj h −j h εj h −j h +1 = εj h −1 εj h , t t • if ih < j h and ih − j h = biht − b jth then εih −bih εih −bih +1 = εih −1 εih , t

t

where indices iht , jth , biht , b jth are given by conditions corresponding to (15) and (18). Theorem 5 Let Pn be the n × n backward identity matrix. A nonsingular matrix A = (aij )1≤i,j≤n is ASSR with signature ε = (ε1 , ε2 , . . . ,εn ), with ε2 = −1 if and only if for every h = 1, . . . , n − 1 the following properties holds simultaneously: (i) B = Pn A is type-I staircase; (ii) the Neville elimination of the matrices Bh = Pn Ah and BhT = Pn ATh can be performed without row exchanges; (iii) the pivots phij of the Neville elimination of Bh satisfy conditions corresponding to (13), h of the Neville elimination of B T satisfy (16) and (17); (14), and the pivots qij h (iv) for the positions (ih , j h ) of matrix Pn Ah : • if ih ≥ j h and ih − j h = iht − jth then εj h −j h εj h −j h +1 = εj h −1 εj h , t t • if ih < j h and ih − j h = biht − b jth then εih −bih εih −bih +1 = εih −1 εih , t

where indices

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iht , jth , biht , b jth

t

are given by conditions corresponding to (15) and (18).

ISBN: 978-84-616-2723-3

Almost Strictly Sign Regular matrices

Acknowledgements This work has been partially supported by the Spanish Research Grant MTM2012-31544 and under MEC and FEDER Grant TEC2012-38142-C04-04.

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