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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 Mn,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 = (aij), we denote by ZF (A) the set of all n × k matrices over F with zero-nonzero pattern A, i.e., ZF (A) = {B = (bij) ∈ Mn,k(F)| bij = 0 if and only if aij = 0}. Thus ∗ indicates nonzero entries.

A matrix with entries from the set {0, ∗} is called a zero-nonzero pattern. We denote by Mn,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 = (aij), we denote by ZF (A) the set of all n × k matrices over F with zero-nonzero pattern A, i.e., ZF (A) = {B = (bij) ∈ Mn,k(F)| bij = 0 if and only if aij = 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 ZF (A) is singular. A zero-nonzero pattern that is not intrinsically singular is called potentially nonsingular.

Given a field F, SIn(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 ∈ ZF (A), B−1 ∈ ZF (A). This notation suggests that the inverse invariant patterns are also called “self-inverse” patterns.

Given a field F, SIn(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 ∈ ZF (A), B−1 ∈ ZF (A). This notation suggests that the inverse invariant patterns are also called “self-inverse” patterns. ITn(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 ∈ ZF (P), Q−1 ∈ ZF (P T ). This notation suggests that the inverse has the transposed pattern. The patterns in ITn(F) are pattern analogs of orthogonal matrices.

Given a field F, SIn(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 ∈ ZF (A), B−1 ∈ ZF (A). This notation suggests that the inverse invariant patterns are also called “self-inverse” patterns. ITn(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 ∈ ZF (P), Q−1 ∈ ZF (P T ). This notation suggests that the inverse has the transposed pattern. The patterns in ITn(F) are pattern analogs of orthogonal matrices. The purpose is to determine the irreducible patterns in SIn(F). To do so, we need first characterize the set ITn(F). The corresponding problems on sign patterns have been studied by

• ther 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 ∈ ITn(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 ∈ ITn(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 F2 = {0, 1}. When n ≥ 4 is even, consider the following zero-nonzero pattern of order n: A =       ∗ · · · ∗ ∗ ... ... . . . . . . ... ... ∗ ∗ · · · ∗       . A ∈ ITn(F2). 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 ∈ SIn(F) if and only if A is one of the following patterns: [∗], ∗ ∗

• ,

∗ ∗ ∗ ∗

• ,

    ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗     ,     ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗     ,     ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗     .

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

• ,

∗ ∗ ∗ ∗

• ,

    ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗     ,     ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗     ,     ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗     . Theorem 2 does not hold for the field F2 = {0, 1}. Consider A =         ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗         ∈ SI6(F2).

• II. Extremal sparsity of the companion

matrix of a polynomial

The companion matrix of a monic polynomial p(x) = xn + a1xn−1 + · · · + an−1x + an

• ver a field is defined to be

C(p) =        · · · −an 1 · · · −an−1 1 · · · −an−2 . . . . . . ... . . . . . . · · · 1 −a1        .

The companion matrix of a monic polynomial p(x) = xn + a1xn−1 + · · · + an−1x + an

• ver a field is defined to be

C(p) =        · · · −an 1 · · · −an−1 1 · · · −an−2 . . . . . . ... . . . . . . · · · 1 −a1        . 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 a1, . . . , an of p(x) as distinct indeterminates, then C(p) has 2n − 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 x1, . . . , xn be distinct indeterminates. We denote by F[x1, . . . , xn] the ring of polynomials in x1, . . . , xn

• ver F, and by F(x1, . . . , xn) the field of rational functions in

x1, . . . , xn over F : F(x1, . . . , xn) = f g

• f, g ∈ F[x1, . . . , xn], g = 0
• .

Let F be a field and x1, . . . , xn be distinct indeterminates. We denote by F[x1, . . . , xn] the ring of polynomials in x1, . . . , xn

• ver F, and by F(x1, . . . , xn) the field of rational functions in

x1, . . . , xn over F : F(x1, . . . , xn) = f g

• f, g ∈ F[x1, . . . , xn], g = 0
• .

Mn(E) : the set of n × n matrices whose entries are elements of a given field E.

Let F be a field and x1, . . . , xn be distinct indeterminates. We denote by F[x1, . . . , xn] the ring of polynomials in x1, . . . , xn

• ver F, and by F(x1, . . . , xn) the field of rational functions in

x1, . . . , xn over F : F(x1, . . . , xn) = f g

• f, g ∈ F[x1, . . . , xn], g = 0
• .

Mn(E) : the set of n × n matrices whose entries are elements of a given field E. Theorem 3 Let F be a field, let a1, . . . , an be distinct indeterminates, and let A ∈ Mn(F(a1, . . . , an)). If the characteristic polynomial of A is xn + a1xn−1 + · · · + an−1x + an, then A has at least 2n − 1 nonzero entries.

Tools from algebra and graph theory in the proof Let F ⊆ K be a field extension. A transcendence basis of K

• ver F is a subset S of K which is algebraically independent
• ver 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

• ver F is a subset S of K which is algebraically independent
• ver 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.

Tools from algebra and graph theory in the proof Let F ⊆ K be a field extension. A transcendence basis of K

• ver F is a subset S of K which is algebraically independent
• ver 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. A branching is an oriented tree having a root of in-degree 0 and all other vertices of in-degree 1. A spanning branching of a digraph is a branching that includes all vertices of the digraph.

Key lemmas Lemma 4 The polynomial xn + a1xn−1 + · · · + an−1x + an is irreducible over F(a1, . . . , an).

Key lemmas Lemma 4 The polynomial xn + a1xn−1 + · · · + an−1x + an is irreducible over F(a1, . . . , an). Lemma 5 In a strongly connected digraph, every vertex is the root of a spanning branching.

Key lemmas Lemma 4 The polynomial xn + a1xn−1 + · · · + an−1x + an is irreducible over F(a1, . . . , an). Lemma 5 In a strongly connected digraph, every vertex is the root of a spanning branching. Lemma 6 Let E be a field. If the digraph of a matrix A ∈ Mn(E) has a spanning branching whose arcs are (i1, j1), . . . , (in−1, jn−1), then there exists a nonsingular diagonal matrix G ∈ Mn(E) such that GAG−1(ik, jk) = 1 for k = 1, . . . , n − 1.

References [1] C. Ma and X. Zhan, Inverse invariant zero-nonzero patterns, Linear Algebra Appl., 443(2014), 184-190. [2] C. Ma and X. Zhan, Extremal sparsity of the companion matrix of a polynomial, Linear Algebra Appl., 438(2013), 621-625.

AMS GSM 147

Matrix Theory Zhan

GSM/147 For additional information and updates on this book, visit www.ams.org/bookpages/gsm-147 www.ams.org AMS on the Web www.ams.org Matrix theory is a classical topic of algebra that had originated, in its current form, in the middle of the 19th century. It is remarkable that for more than 150 years it continues to be an active area of research full of new discoveries and new applications. This book presents modern perspectives of matrix theory at the level accessible to graduate students. It differs from other books on the subject in several aspects. First, the book treats certain topics that are not found in the standard textbooks, such as completion of partial matrices, sign patterns, applications of matrices in combinatorics, number theory, algebra, geometry, and polynomials. There is an appendix of unsolved problems with their history and current state. Second, there is some new material within traditional topics such as Hopf’s eigenvalue bound for positive matrices with a proof, a proof of Horn’s theorem on the converse of Weyl’s theorem, a proof of Camion-Hoffman’s theorem on the converse of the diagonal dominance theorem, and Audenaert’s elegant proof of a norm inequality for commutators. Third, by using powerful tools such as the compound matrix and Gröbner bases of an ideal, much more concise and illuminating proofs are given for some previously known results. This makes it easier for the reader to gain basic knowledge in matrix theory and to learn about recent developments.

264 pages • backspace 1 1/4 • 3-color cover PMS 632 (light blue), PMS 302(blue), 123 (yellow)

American Mathematical Society

Xingzhi Zhan

Matrix Theory

Graduate Studies in Mathematics

Volume 147

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