Spaces of matrices of equal rank

Let M_m,n(F) denote the space of all mXn matrices over the algebraically closed field F. A subspace of M_m,n(F), all of whose nonzero elements have rank k, is said to be essentially decomposable if there exist nonsingular mXn matrices U and V respectively such that for any element A, UAV has the form

$\text{UAV=}\text{A}{}_1\text{A}{}_2\text{A}{}_3\text{0}$

where A₁ is iX(k–i) for some i?k. Theorem: If $\text{K}$ is a space of rank k matrices, then either $\text{K}$ is essentially decomposable or dim $\text{K}$?k+1. An example shows that the above bound on non-essentially-decomposable spaces of rank k matrices is sharp whenever n?2k–1.     

Products of reflections in the general linear group over a division ring

Let F be a division ring and A?GL_n(F). We determine the smallest integer k such that A admits a factorization A=R₁R₂?R_k?1B, where R₁,…,R_k?1 are reflections and B is such that rank(B?I_n)=1. We find that, apart from two very special exceptional cases, k=rank(A?I_n). In the exceptional cases k is one larger than this rank. The first exceptional case is the matrices A of the form I_m⊕αI_n?m where n?m?2, α≠?1, and α belongs to the center of F. The second exceptional case is the matrices A satisfying (A?I_n)²=0, rank(A?I_n)?2 in the case when char F≠2 only. This result is used to determine, in the case when F is commutative, the length of a matrix A?GL_n(F) with detA=±1 with respect to the set of all reflections in GL_n(F).     

A generalization of Lavoie's inequality concerning the sum of idempotent matrices

Let a complex pxn matrix A be partitioned as A′=(A′₁,A′₂,…,A′_k). Denote by ?(A), A′, and A^? respectively the rank of A, the transpose of A, and an inner inverse (or a g-inverse) of A. Let A⁽¹⁴⁾ be an inner inverse of A such that A⁽¹⁴⁾A is a Hermitian matrix. Let B=(A⁽¹⁴⁾₁,A⁽¹⁴⁾₂,…,A_k⁽¹⁴⁾) and $\text{ρ(A)=}\text{∑}\text{i=1}\text{k}\text{ρ(A}{}_{\mathrm{i}}\text{)}$.Then the product of nonzero eigenvalues of BA (or AB) cannot exceed one, and the product of nonzero eigenvalues of BA is equal to one if and only if either B=A⁽¹⁴⁾ or $\text{A}{}_{\mathrm{i>}}\text{A}{}_{\mathrm{<mtext>j</mtext>}}{}^1\text{=0}$ for all i ≠ j,i, j=1,2,…,k . The results of Lavoie (1980) and Styan (1981) are obtained as particular cases. A result is obtained for k=2 when the condition $\text{ρ(A)=}\text{∑}\text{i=1}\text{k}\text{ρ(A}{}_{\mathrm{i}}\text{)}$ is no longer true.     

The rank of a difference of matrices and associated generalized inverses

Various representations are given to characterize the rank of A-S in terms of rank A+k where A and S are arbitrary complex matrices and k is a function of A and S. It is shown that if S=AMA for some matrix M, and if G is any matrix satisfying A=AGA, then

$\text{rank(A-S) = rankA-nullity (I-SG)}$

. Several alternative forms of this result are established, as are many equivalent conditions to have

$\text{rank(A-S) = rankA-rankS}$

. General forms for the Moore-Penrose inverse of matrices A-S are developed which include as special cases various results by Penrose, Wedin, Hartwig and others.     

Involutory matrices over finite commutative rings

An n × n matrix A is called involutory iff A²=I_n, where I_n is the n × n identity matrix. This paper is concerned with involutory matrices over an arbitrary finite commutative ring R with identity and with the similarity relation among such matrices. In particular the authors seek a canonical set $\text{C}$ with respect to similarity for the n × n involutory matrices over R—i.e., a set $\text{C}$ of n × n involutory matrices over R with the property that each n × n involutory matrix over R is similar to exactly on matrix in $\text{C}$. Because of the structure of finite commutative rings and because of previous research, they are able to restrict their attention to finite local rings of characteristic a power of 2, and although their main result does not completely specify a canonical set $\text{C}$ for such a ring, it does solve the problem for a special class of rings and shows that a solution to the general case necessarily contains a solution to the classically unsolved problem of simultaneously bringing a sequence A₁,…,A_v of (not necessarily involutory) matrices over a finite field of characteristic 2 to canonical form (using the same similarity transformation on each A_i). (More generally, the authors observe that a theory of similarity fot matrices over an arbitrary local ring, such as the well-known rational canonical theory for matrices over a field, necessarily implies a solution to the simultaneous canonical form problem for matrices over a field.) In a final section they apply their results to find a canonical set for the involutory matrices over the ring of integers modulo 2^m and using this canonical set they are able to obtain a formula for the number of n × n involutory matrices over this ring.     

Commutative algebras of minimal rank

If $\text{A}$ is a finite-dimensional associative k-algebra with unit, then its rank R($\text{A}$), i.e. its bilinear complexity, is never less than $\text{2dim}\text{A}\text{?\#}\text{M}\text{(}\text{A}\text{)}$. $\text{A}$ is said to be of minimal rank if $\text{R(}\text{A}\text{)=2dim}\text{A}\text{?\#}\text{M}\text{(}\text{A}\text{)}$. In this paper we determine for infinite perfect fields k all commutative k-algebras of minimal rank. Roughly speaking, these algebras are built up from simply generated structures which annihilate each other. Furthermore, we indicate how this result can be used to obtain new lower bounds for the rank of specific commutative algebras.     

Restoring rank and consistency by orthogonal projection

Let A₀x=b₀ be a consistent (but possibly unknown) linear algebraic system of m equations in n unknowns, with rank(A₀)=k (likewise possibly unknown). Let Ax=b be a known (but possibly inconsistent) nearby system. Procedures for “solving” Ax=b usually replace it (at least in principle) by a nearby consistent system $\text{A}\text{?}\text{x=}\text{b}\text{?}$ of (hopefully) rank k, and solve that one instead. We consider consistent systems $\text{A}\text{?}\text{x=}\text{b}\text{?}$ with rank$\text{(}\text{A}\text{?}\text{)=k}$ such that (Ã,b?) is the orthogonal projection of (A,b) on span(Ã) [i.e., the columns of (Ã┆b?) are the projections of those of (A ┆ b)]. Under suitable circumstances the rank k pair (Ã,b?) nearest to (A,b) in the sense of minimizing $\text{6(}\text{A}\text{?}\text{┆}\text{b}\text{?}\text{) ? (}\text{A}\text{?}\text{┆}\text{b}\text{?}\text{)6}{}_{\mathrm{F}}$ belongs to this class, as well as the pair delivered by the ordinary least squares method (if k=n?m) or, more generally, the pair delivered by a well-known algorithm (viz. HFTI). If $\text{?:= |(A┆b)?(A}{}_0\text{┆b}{}_0\text{)6}{}_{\mathrm{F}}$, then the minimum length solutions of all systems $\text{A}\text{?}\text{x=}\text{b}\text{?}$ so related to (A,b) are shown to differ mutually only by O(?²). This means, e.g., that itwill usually not pay to compute the solution of the nearest system.     

Determinants of Galois automorphisms of maximal commutative rings of 2×2 matrices

The determinants of solutions X to any of the 2×2 matrix equations: (1) XAX>^?1=A^t, t denoting transpose; (2) A=XY, X, Y symmetric; (3) X=AB?BA; and (4) $\text{XAX}{}^{\mathrm{?1}}\text{=A}{}^1$, the matrix of cofactors of A transposed, are characterized over a commutative ring R as the negative of norms from the quadratic extension R[A] whenever certain elements in the Picard group Pic (R[A]) are trivial. This characterization is realized by the utilization of galois cohomology and a generalized Latimer–MacDuffee correspondence between similarity classes and the elements in Pic (R[A]). The results represent a ramification of several articles by O. Taussky on the specializations of equations (1)–(3) to either the field of rational numbers or the ring of rational integers.     

Generalized inverse-positivity and splittings of M-matrices

The concepts of matrix monotonicity, generalized inverse-positivity and splittings are investigated and are used to characterize the class of all M-matrices A, extending the well-known property that A^?1?0 whenever A is nonsingular. These conditions are grouped into classes in order to identify those that are equivalent for arbitrary real matrices A. It is shown how the nonnegativity of a generalized left inverse of A plays a fundamental role in such characterizations, thereby extending recent work by one of the authors, by Meyer and Stadelmaier and by Rothblum. In addition, new characterizations are provided for the class of M-matrices with “property c”; that is, matrices A having a representation A=sI?B, s>0, B?0, where the powers of $\text{(}\text{1}\text{s}\text{)B}$ converge. Applications of these results to the study of iterative methods for solving arbitrary systems of linear equations are given elsewhere.     

Preservers of pairs of bivectors with bounded distance

We extend Liu’s fundamental theorem of the geometry of alternate matrices to the second exterior power of an infinite dimensional vector space and also use her theorem to characterize surjective mappings T from the vector space V of all n×n alternate matrices over a field with at least three elements onto itself such that for any pair A, B in V, rank(A-B)?2k if and only if rank(T(A)-T(B))?2k, where k is a fixed positive integer such that n?2k+2 and k?2.     

On the matrix monotonicity of generalized inversion

Let A, B be two matrices of the same order. We write A>B(A>?B) iff A? B is a positive (semi-) definite hermitian matrix. In this paper the well-known result if

$\text{A>B>θ, then B}{}^{\mathrm{?1}}\text{>A}{}^{\mathrm{?1}}\text{> θ}$

(cf. Bellman [1, p.59]) is extended to the generalized inverses of certain types of pairs of singular matrices A,B?θ, where θ denotes the zero matrix of appropriate order.     

Embedding of a pseudo-point residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let υ, t, λ and k be nonnegative integers such that υ?k?t?0. A pair (X, $\text{U}$) is called a (υ, k, λ) t-design, denoted by S_λ(t, k, υ), if (1) |X| = υ, (2) every t-subset of X is contained in exactly λ blocks and (3) for every block A in $\text{U}$, |A| = k. A Möbius plane M is an S₁(3, q+1, q²+1) where q is a positive integer. Let ∞ be a fixed point in M. If ∞ is deleted from M, together with all the blocks containing ∞, then we obtain a point-residual design M^*. It can be easily checked that M^* is an S_q(2, q+1, q²). Any S_q(2, q+1, q²) is called a pseudo-point-residual design of order q, abbreviated by PPRD(q). Let A and B be two blocks in a PPRD(q)M^*. A and B are said to be tangent to each other at z if and only if A∩B={z}. M^* is said to have the Tangency Property if for any block A in M^*, and points x and y such that x?A and y?A, there exists at most one block containing y and tangent to A at x. This paper proves that any PPRD(q)M^* is uniquely embeddable into a Möbius plane if and only if M^* satisfies the Tangency Property.     

Simple product colorings

If A is a set colored with m colors, and B is colored with n colors, the coloring of A × B obtained by coloring (a, b) with the pair (color of a, color of b) will be called an m × n simple product coloring (SPC) of A × B. SPC's of Cartesian products of three or more sets are defined analogously. It is shown that there are 2 × 2, and 2 × 2 × 2 SPC's of $\text{Q}$² and $\text{Q}$³ which forbid the distance one; that there is no 2^k SPC of $\text{Q}$^k forbidding the distance one, for k > 3; and that there is no 2 × 2 SPC of $\text{Q}$ × $\text{Q}$(√15), and thus none of $\text{R}$², forbidding the distance 1.     

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.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

On matrices having equal corresponding principal minors

Let A, B be n × n matrices with entries in a field F. We say A and B satisfy property $\text{D}$ if B or B^t is diagonally similar to A. It is clear that if A and B satisfy property $\text{D}$, then they have equal corresponding principal minors, of all orders. The question is to what extent the converse is true. There are examples which show the converse is not always true. We modify the problem slightly and give conditions on a matrix A which guarantee that if B is any matrix which has the same principal minors as A, then A and B will satisfy property $\text{D}$. These conditions on A are formulated in terms of ranks of certain submatrices of A and the concept of irreducibility.     

On the convergence of powers of interval matrices

We give a necessary and sufficient condition for the sequence {$\text{A}$^k}of the powers of an interval matrix $\text{A}$ to converge to the null matrix $\text{O}$. We construct a point matrix $\text{B}$ which has spectral radius ? ($\text{B}$) less than one if {$\text{A}$^k}converges.     

A note on k-plane integral transforms

Let Π be a k-dimensional subspace of Rⁿ, n ? 2, and write x = (x′, x″) with x′ in Π and x″ in the orthogonal complement Π^⊥. The k-plane transform of a measurable function ? in the direction Π at the point x″ is defined by $\text{L?(Π, x″) = ∝Π?(x′, x″) dx′}$. In this article certain a priori inequalities are established which show in particular that if $\text{? ? L}{}^{\mathrm{p}}\text{(R}{}^{\mathrm{n}}\text{)}$, $\text{1 ? p}\text{\$}\text{?}\text{n}\text{k}$, then ? is integrable over almost every translate of almost every k-space. Mapping properties of the k-plane transform between the spaces L^p(Rⁿ), p ? 2, and certain Lebesgue spaces with mixed norm on a vector bundle over the Grassmann manifold of k-spaces in Rⁿ are also obtained.     

Lower bounds for the norm of the symmetric product

Suppose each of m, n, and k is a positive integer, k ? n, A is a (real-valued) symmetric n-linear function on E_m, and B is a k-linear symmetric function on E_m. The tensor and symmetric products of A and B are denoted, respectively, by A ?B and A?B. The identity

$\text{6A · B6}{}^2\text{=}\text{∑}\text{q=0}\text{n}\text{(}\text{n}\text{k}\text{)}\text{(}\text{n+k}\text{k}\text{)}\text{6A?}{}_{\mathrm{q}}\text{B6}{}^2$

is proven by Neuberger in [1]. An immediate consequence of this identity is the inequality

$\text{6A · B 6}{}^2\text{?}\text{n+k}\text{n}{}^{\mathrm{?1}}\text{6A · B 6}{}^2$

In this paper a necessary and sufficient condition for

$\text{6A · B 6}{}^2\text{=}\text{n+k}\text{n}{}^{\mathrm{?}}\text{6A · B 6}{}^2$

is given. It is also shown that under certain conditions the inequality can be considerably improved. This improvement results from an analysis of the terms 6A?_qB6, 1?q?n, appearing in the identity.