Generalized inverses of matrices over fields of characteristic two

For F a field of characteristic two, the problem of determining which m×n matrices of rank r have normalized generalized inverses and which have pseudoinverses is solved. For F_q a finite field of characteristic two, both the number of m×n matrices of rank r over F which have normalized generalized inverses and the number of m×n matrices of rank r over F_q which have pseudoinverses are determined.     

M-P invertible matrices and unitary groups over Fq 2

The Moor-Penrose generalized inverses (M-P inverses for short) of matrices over a finite field F_q ² which is a generalization of the Moor-Penrose generalized inverses over the complex field, are studied in the present paper. Some necessary and sufficient conditions for anm xn matrixA over F_q ² having an M-P inverse are obtained, which make clear the set ofm xn matrices over F_q ² having M-P inverses and reduce the problem of constructing and enumerating the M-P invertible matrices to that of constructing and enumerating the non-isotropic subspaces with respect to the unitary group. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing the results in geometry of unitary groups over finite fields.     

Enumeration of matrices of given rank with submatrices of given rank

The authors determine the number of (n+m)×t matrices A¹ of rank r+v, over a finite field GF(q), whose last m rows are those of a given matrix A of rank r+v over GF(q) and whose first n rows have a given rank u.     

M-P invertible matrices and unitary groups over Fq2

The Moor-Penrose generalized inverses (M-P inverses for short) of matrices over a finite field F_q ² which is a generalization of the Moor-Penrose generalized inverses over the complex field, are studied in the present paper. Some necessary and sufficient conditions for anm xn matrixA over F_q ² having an M-P inverse are obtained, which make clear the set ofm xn matrices over F_q ² having M-P inverses and reduce the problem of constructing and enumerating the M-P invertible matrices to that of constructing and enumerating the non-isotropic subspaces with respect to the unitary group. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing the results in geometry of unitary groups over finite fields.     

Group inverse for two classes of 2×2 anti-triangular block matrices over right Ore domains

Suppose ? is a right Ore domain with identity 1. Let ? ^m×n denote the set of all m×n matrices over ?. In this paper, we give the necessary and sufficient conditions for the existence and explicit representations of the group inverses of the block matrices in the following two cases, respectively:

1. A ²=A and CA=C;
2. r(C)≤r(B) and A=DC,

where A∈? ^n×n , B∈? ^n×m , C∈? ^m×n , D∈? ^n×m . The paper’s results generalize some relative results of Liu and Yang (Appl. Math. Comput. 218:8978–8986, 2012).     

Possible isolation number of a matrix over nonnegative integers

Let ?₊ be the semiring of all nonnegative integers and A an m × n matrix over ?₊. The rank of A is the smallest k such that A can be factored as an m × k matrix times a k×n matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of A is a lower bound of the rank of A. For A with isolation number k, we investigate the possible values of the rank of A and the Boolean rank of the support of A. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is 1 or 2 only. We also determine a special type of m×n matrices whose isolation number is m. That is, those matrices are permutationally equivalent to a matrix A whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.     

The nonnegative rank factorizations of nonnegative matrices

Let A ∈ P^{m × n}_r, the set of all m × n nonnegative matrices having the same rank r. For matrices A in P^{m × n}_n, we introduce the concepts of “A has only trivial nonnegative rank factorizations” and “A can have nontrivial nonnegative rank factorizations.” Correspondingly, the set P^{m × n}_n is divided into two disjoint subsets P₍₁₎ and P₍₂₎ such that P₍₁₎∪P₍₂₎ = P^{m × n}_n. It happens that the concept of “A has only trivial nonnegative rank factorizations” is a generalization of “A is prime in P^{n × n}_n.” We characterize the sets P₍₁₎ and P₍₂₎. Some of our results generalize some theorems in the paper of Daniel J. Richman and Hans Schneider [9].     

A linear-algebra problem from algebraic coding theory

Given an m×n matrix M over E=GF(q^t) and an ordered basis A={z₁,…,z_t} for field E over K=GF(q), expand each entry of M into a t×1 vector of coordinates of this entry relative to A to obtain an mt×n matrix M¹ with entries from the field K. Let r=rank(M) and r¹=rank(M¹). We show that r?r¹?min{rt,n}, and we determine the number b(m,n,r,r¹,q,t) of m×n matrices M of rank r over GF(q^t) with associated mt×n matrix M¹ of rank r¹ over GF (q).     

Linear transformations which preserve fixed rank

Let M_{m, n}(F) denote the set of all m×n matrices over the algebraically closed field F. Let T denote a linear transformation, T:M_{m, n}(F)→M_{m, n}(F). Theorem: If max(m, n)?2k?2, k?1, and T preserves rank k matrices [i.e.?(A)=k implies ?(T(A))=k], then there exist nonsingular m×m and n×n matrices U and V respectively such that either (i) T:A→UAV for all A?M_{m, n}(F), or (ii) m=n and T:A→UA^tV for all A?M_n(F), where A^t denotes the transpose of A.     

On generalized inverses of matrices over integral domains

It is proved that a matrix A over an integral domain admits a 1-inverse if and only if a linear combination of all the r × r minors of A is equal to one, where r is the rank of A. Some results on the existence of Moore-Penrose inverses are also obtained.     

On the plane term rank of three dimensional matrices

It is shown that if A is a p>×q×r matrix such that each of the horizontal plane sections of A has full term rank, then the plane term rank of A is greater than m?√m where m= min {p,q,r}. In particular, if A is a three dimensional line stochastic matrix of order n, then the plane term rank of A is greater than n?√n.     

On distribution by rank of bases for vector spaces of matrices over a finite field

Let F^m×n_q denote the vector space of all m×n matrices over the finite field F_q of order q, and let $\text{B}\text{=(A}{}_1\text{,A}{}_2\text{,…,A}{}_{\mathrm{mn}}\text{)}$ denote an ordered basis for F^m×n_q. If the rank of A_i is r_i,i=1,2,…,mn, then $\text{B}$ is said to have rank (r₁,r₂,…,r_mn), and the number of ordered bases of F^mxn_q with rank (r₁,r₂,…,r_mn is denoted by N_q(r₁, r₂,…,r_mn). This paper determines formulas for the numbers N_q(r₁,r₂,…,r_mn) for the case m=n=2, q arbitrary, and while some of the techniques of the paper extend to arbitrary m and n, the general formulas for the numbers N_q(r₁,r₂,…,r_mn) seem quite complicated and remain unknown. An idea on a possible computer attack which may be feasible for low values of m and n is also discussed.     

The Moore-Penrose inverse of a morphism with factorization

Given an m-by-n matrix A of rank r over a field with an involutory automorphism, it is well known that A has a Moore-Penrose inverse if and only if rank A¹A=r= rank AA¹. By use of the full-rank factorization theorem, this result may be restated in the category of finite matrices as follows: if (A₁, r, A₂) is an (epic, monic) factorization of A:m→n through r, then A has a Moore-Penrose inverse if and only if (A¹A₁, r, A₂) and (A₁, r, A₂A¹) are, respectively, (epic, monic) factorizations of A¹A:n→n and AA¹:m→m through r. This characterization of the existence of Moore-Penrose inverses is extended to arbitrary morphisms with (epic, monic) factorizations.     

Solvability of the Matric Equation AX = B

Let A be an n×s matrix of rank r, B be an n×t matrix of rank ρ?r, and X be an s×t matrix. This paper discusses conditions on the matrices A and B so that the matric equation AX=B will have solutions for the matrix X.     

Partition,Construction, and Enumeration of M–P Invertible Matrices over Finite Fields

A necessary and sufficient condition for an m×n matrix A over F_q having a Moor–Penrose generalized inverse (M–P inverse for short) was given in (C. K. Wu and E. Dawson, 1998, Finite Fields Appl. 4, 307–315). In the present paper further necessary and sufficient conditions are obtained, which make clear the set of m×n matrices over F_q having an M–P inverse and reduce the problem of constructing M–P invertible matrices to that of constructing subspaces of certain type with respect to some classical groups. Moreover, an explicit formula for the M–P inverse of a matrix which is M–P invertible is also given. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing results in geometry of classical groups over finite fields (Z. X. Wan, 1993, “Geometry of Classical Groups over Finite Fields”, Studentlitteratur, Chatwell Bratt).     

A comparison of nonnegative real ranks and their preservers

This paper concerns three notions of rank of matrices over semirings; real rank, semiring rank and column rank. These three rank functions are the same over subfields of reals but differ for matrices over subsemirings of nonnegative reals. We investigate the largest values of r for which the real rank and semiring rank, real rank and column rank of all m×n matrices over a given semiring are both r, respectively. We also characterize the linear operators which preserve the column rank of matrices over certain subsemirings of the nonnegative reals.     

Rank-k-preservers and preservers of sets of ranks

Let T be a linear transformation on the set of m × n matrices with entries in an algebraically closed field. If T maps the set of all matrices whose rank is k into itself, and if$\text{n?}\text{3}\text{k}\text{2}$, then the rank of A is the rank of T(A) for every m × n matrix.     

Forward order law for the generalized inverses of multiple matrix product

The generalized inverses have many important applications in the aspects of theoretic research and numerical computations and therefore they were studied by many authors. In this paper we get some necessary and sufficient conditions of the forward order law for 1-inverse of multiple matrices productsA =A ₁ A ₂ … A _n by using the maximal rank of generalized Schur complement.     

Linear maps preserving M–P inverses of matrices between matrix spaces over fields

Suppose F is a field of characteristic not 2. Let n and m be two arbitrary positive integers with n≥2. We denote by M _n (F) and S _n (F) the space of n×n full matrices and the space of n×n symmetric matrices over F, respectively. All linear maps from S _n (F) to M _m (F) preserving M–P inverses of matrices are characterized first, and thereby all linear maps from S _n (F) (M _n (F)) to S _m (F) (M _m (F)) preserving M–P inverses of matrices are characterized, respectively.     

Products of idempotent matrices

For some years it has been known that every singular square matrix over an arbitrary field F is a product of idempotent matrices over F. This paper quantifies that result to some extent. Main result: for every field F and every pair (n,k) of positive integers, an n×n matrix S over F is a product of k idempotent matrices over F iff rank(I ? S)?k· nullity S. The proof of the “if” part involves only elementary matrix operations and may thus be regarded as constructive. Corollary: (for every field F and every positive integer n) each singular n×n matrix over F is a product of n idempotent matrices over F, and there is a singular n×n matrix over F which is not a product of n ? 1 idempotent matrices.