Inverse invariant zero-nonzero patterns Xingzhi Zhan - PowerPoint PPT Presentation

Inverse invariant zero-nonzero patterns Xingzhi Zhan zhan@math.ecnu.edu.cn East China Normal University Joint work with Chao Ma

A matrix with entries from the set { 0 , ∗} is called a zero-nonzero pattern. We denote by M n,k ( F ) the set of all n × k matrices over a field F. Given a field F and an n × k zero-nonzero pattern A = ( a ij ), we denote by Z F ( A ) the set of all n × k matrices over F with zero-nonzero pattern A , i.e., Z F ( A ) = { B = ( b ij ) ∈ M n,k ( F ) | b ij = 0 if and only if a ij = 0 } . Thus ∗ indicates nonzero entries.

A matrix with entries from the set { 0 , ∗} is called a zero-nonzero pattern. We denote by M n,k ( F ) the set of all n × k matrices over a field F. Given a field F and an n × k zero-nonzero pattern A = ( a ij ), we denote by Z F ( A ) the set of all n × k matrices over F with zero-nonzero pattern A , i.e., Z F ( A ) = { B = ( b ij ) ∈ M n,k ( F ) | b ij = 0 if and only if a ij = 0 } . Thus ∗ indicates nonzero entries. A square zero-nonzero pattern A is said to be intrinsically singular over a field F if every matrix in Z F ( A ) is singular. A zero-nonzero pattern that is not intrinsically singular is called potentially nonsingular.

Given a field F , SI n ( F ) denotes the set of all zero-nonzero patterns A of order n which are potentially nonsingular and satisfy the condition that for every nonsingular matrix B ∈ Z F ( A ) , B − 1 ∈ Z F ( A ) . This notation suggests that the inverse invariant patterns are also called “self-inverse” patterns.

Given a field F , SI n ( F ) denotes the set of all zero-nonzero patterns A of order n which are potentially nonsingular and satisfy the condition that for every nonsingular matrix B ∈ Z F ( A ) , B − 1 ∈ Z F ( A ) . This notation suggests that the inverse invariant patterns are also called “self-inverse” patterns. IT n ( F ) denotes the set of all zero-nonzero patterns P of order n which are potentially nonsingular and satisfy the condition that for every nonsingular matrix Q ∈ Z F ( P ) , Q − 1 ∈ Z F ( P T ) . This notation suggests that the inverse has the transposed pattern. The patterns in IT n ( F ) are pattern analogs of orthogonal matrices.

Given a field F , SI n ( F ) denotes the set of all zero-nonzero patterns A of order n which are potentially nonsingular and satisfy the condition that for every nonsingular matrix B ∈ Z F ( A ) , B − 1 ∈ Z F ( A ) . This notation suggests that the inverse invariant patterns are also called “self-inverse” patterns. IT n ( F ) denotes the set of all zero-nonzero patterns P of order n which are potentially nonsingular and satisfy the condition that for every nonsingular matrix Q ∈ Z F ( P ) , Q − 1 ∈ Z F ( P T ) . This notation suggests that the inverse has the transposed pattern. The patterns in IT n ( F ) are pattern analogs of orthogonal matrices. The purpose is to determine the irreducible patterns in SI n ( F ) . To do so, we need first characterize the set IT n ( F ) . The corresponding problems on sign patterns have been studied by other authors and their results can be deduced by the results here. Note that sign patterns are special (more precise) zero-nonzero patterns.

Theorem 1 Let F be a field with | F | ≥ 3 and let A be a zero-nonzero pattern of order n . Then A ∈ IT n ( F ) if and only if A is permutation equivalent to a block diagonal matrix with � ∗ � ∗ each diagonal block being [ ∗ ] or . ∗ ∗

Theorem 1 Let F be a field with | F | ≥ 3 and let A be a zero-nonzero pattern of order n . Then A ∈ IT n ( F ) if and only if A is permutation equivalent to a block diagonal matrix with � ∗ � ∗ each diagonal block being [ ∗ ] or . ∗ ∗ Theorem 1 does not hold for the field F 2 = { 0 , 1 } . When n ≥ 4 is even, consider the following zero-nonzero pattern of order n :   0 ∗ · · · ∗ . ... ... .   ∗ .   A = .  .  ... ... .   . ∗   ∗ · · · ∗ 0 A ∈ IT n ( F 2 ). But A can not be permutation equivalent to a block diagonal matrix with each diagonal block being [ ∗ ] or � ∗ � ∗ . This can be seen by considering the number of zero ∗ ∗ entries.

Theorem 2 Let F be a field with | F | ≥ 3 and let A be an irreducible zero-nonzero pattern of order n . Then A ∈ SI n ( F ) if and only if A is one of the following patterns: � 0 � ∗ � � ∗ ∗ [ ∗ ] , , , ∗ 0 ∗ ∗  0 ∗ 0 ∗   0 0 ∗ ∗   0 ∗ ∗ 0  ∗ 0 ∗ 0 0 0 ∗ ∗ ∗ 0 0 ∗        ,  ,  .       0 ∗ 0 ∗ ∗ ∗ 0 0 ∗ 0 0 ∗    ∗ 0 ∗ 0 ∗ ∗ 0 0 0 ∗ ∗ 0

Theorem 2 Let F be a field with | F | ≥ 3 and let A be an irreducible zero-nonzero pattern of order n . Then A ∈ SI n ( F ) if and only if A is one of the following patterns: � 0 � ∗ � � ∗ ∗ [ ∗ ] , , , ∗ 0 ∗ ∗  0 ∗ 0 ∗   0 0 ∗ ∗   0 ∗ ∗ 0  ∗ 0 ∗ 0 0 0 ∗ ∗ ∗ 0 0 ∗        ,  ,  .       0 ∗ 0 ∗ ∗ ∗ 0 0 ∗ 0 0 ∗    ∗ 0 ∗ 0 ∗ ∗ 0 0 0 ∗ ∗ 0 Theorem 2 does not hold for the field F 2 = { 0 , 1 } . Consider  0 ∗ ∗ 0 ∗ 0  0 0 ∗ ∗ 0 ∗     ∗ 0 0 ∗ 0 0   A = ∈ SI 6 ( F 2 ) .   0 ∗ 0 0 ∗ 0     0 0 ∗ 0 0 ∗   ∗ 0 0 ∗ ∗ 0

II. Extremal sparsity of the companion matrix of a polynomial

The companion matrix of a monic polynomial p ( x ) = x n + a 1 x n − 1 + · · · + a n − 1 x + a n over a field is defined to be   0 0 · · · 0 − a n 1 0 · · · 0 − a n − 1     0 1 · · · 0 − a n − 2 C ( p ) = .    . . . .  ... . . . .   . . . .   0 0 · · · 1 − a 1

The companion matrix of a monic polynomial p ( x ) = x n + a 1 x n − 1 + · · · + a n − 1 x + a n over a field is defined to be   0 0 · · · 0 − a n 1 0 · · · 0 − a n − 1     0 1 · · · 0 − a n − 2 C ( p ) = .    . . . .  ... . . . .   . . . .   0 0 · · · 1 − a 1 It is well known that the characteristic polynomial of C ( p ) is p ( x ) .

Because of this relation, companion matrices can be used to 1) locate the roots of a complex polynomial, 2) prove constructively that the algebraic numbers form a field, 3) prove constructively that the algebraic integers form a ring.

Because of this relation, companion matrices can be used to 1) locate the roots of a complex polynomial, 2) prove constructively that the algebraic numbers form a field, 3) prove constructively that the algebraic integers form a ring. The companion matrix C ( p ) is very sparse, i.e., it has many zero entries. If we regard the coefficients a 1 , . . . , a n of p ( x ) as distinct indeterminates, then C ( p ) has 2 n − 1 nonzero entries. We will show that the companion matrix is the sparsest in a sense to be described below.

Let F be a field and x 1 , . . . , x n be distinct indeterminates. We denote by F [ x 1 , . . . , x n ] the ring of polynomials in x 1 , . . . , x n over F, and by F ( x 1 , . . . , x n ) the field of rational functions in x 1 , . . . , x n over F : � f � � � F ( x 1 , . . . , x n ) = � f, g ∈ F [ x 1 , . . . , x n ] , g � = 0 . � g

Let F be a field and x 1 , . . . , x n be distinct indeterminates. We denote by F [ x 1 , . . . , x n ] the ring of polynomials in x 1 , . . . , x n over F, and by F ( x 1 , . . . , x n ) the field of rational functions in x 1 , . . . , x n over F : � f � � � F ( x 1 , . . . , x n ) = � f, g ∈ F [ x 1 , . . . , x n ] , g � = 0 . � g M n ( E ) : the set of n × n matrices whose entries are elements of a given field E.

Let F be a field and x 1 , . . . , x n be distinct indeterminates. We denote by F [ x 1 , . . . , x n ] the ring of polynomials in x 1 , . . . , x n over F, and by F ( x 1 , . . . , x n ) the field of rational functions in x 1 , . . . , x n over F : � f � � � F ( x 1 , . . . , x n ) = � f, g ∈ F [ x 1 , . . . , x n ] , g � = 0 . � g M n ( E ) : the set of n × n matrices whose entries are elements of a given field E. Theorem 3 Let F be a field, let a 1 , . . . , a n be distinct indeterminates, and let A ∈ M n ( F ( a 1 , . . . , a n )) . If the characteristic polynomial of A is x n + a 1 x n − 1 + · · · + a n − 1 x + a n , then A has at least 2 n − 1 nonzero entries.

Tools from algebra and graph theory in the proof Let F ⊆ K be a field extension. A transcendence basis of K over F is a subset S of K which is algebraically independent over F and is maximal with respect to set-theoretic inclusion in the set of all algebraically independent subsets of K.

Tools from algebra and graph theory in the proof Let F ⊆ K be a field extension. A transcendence basis of K over F is a subset S of K which is algebraically independent over F and is maximal with respect to set-theoretic inclusion in the set of all algebraically independent subsets of K. The transcendence degree of K over F is the cardinality of any transcendence basis of K over F.