Line Insertions in Totally Positive Matrices

It is obvious that between any two rows (columns) of an m-by-n totally nonnegative matrix a new row (column) may be inserted to form an (m+1)-by-n (m-by-(n+1)) totally nonnegative matrix. The analogous question, in which “totally nonnegative” is replaced by “totally positive” arises, for example, in completion problems and in extension of collocation matrices, and its answer is not obvious. Here, the totally positive case is answered affirmatively, and in the process an analysis of totally positive linear systems, that may be of independent interest, is used.     

Reachability of condensed forms via unitary congruence transformations

There are several well-known facts about unitary similarity transformations of complex n-by-n matrices: every matrix of order n = 3 can be brought to tridiagonal form by a unitary similarity transformation; if n ≥ 5, then there exist matrices that cannot be brought to tridiagonal form by a unitary similarity transformation; for any fixed set of positions (pattern) S whose cardinality exceeds n(n ? 1)/2, there exists an n-by-n matrix A such that none of the matrices that are unitarily similar to A can have zeros in all of the positions in S. It is shown that analogous facts are valid if unitary similarity transformations are replaced by unitary congruence ones.     

Completion of partial matrices to contractions

An n-by-m partially specified complex matrix is called a partial contraction if every rectangular submatrix consisting entirely of specified entries is itself a contraction. Necessary and sufficient condition are given for the pattern of specified entries such that any n-by-m partial contraction with this pattern may be completed to a full n-by-m contraction.     

The number of equivalence classes of symmetric sign patterns

This paper shows that the number of sign patterns of totally non-zero symmetric n-by-n matrices, up to conjugation by permutation and signature matrices and negation, is equal to the number of unlabelled graphs on n vertices.     

Reconstructing Matrices from Minors

Which collections of mn minors of an m-by-n matrix uniquely determine the matrix, given some regularity conditions? For m=n=3, the 585 such collections, that are distinct up to symmetry, are determined. For general m, n, a necessary and a sufficient condition for reconstruction are given in terms of matchings in a bipartite graph. Among other particular results, those collections of entries for which there are minors that permit reconstruction one entry at a time are characterized.     

On the reduction of pairs of hermitian or symmetric matrices to diagonal form by congruence

Let A₁, A₂ be given n-by-n Hermitian or symmetric matrices, and consider the simultaneous transformations A_i→SA_iS^* if A_i is Hermitian or A_i→SA_iS^T if A_i is symmetric. We give necessary and sufficient conditions for the existence of a unitary S which reduces both A₁ and A₂ to diagonal form in this way. When at least one of A₁ or A₂ is nonsingular, we give necessary and sufficient conditions for a reduction of this sort with a nonsingular S. These results are a generalization of the classical theorem from mechanics that a positive definite matrix and a Hermitian matrix can always be diagonalized simultaneously by a nonsingular congruence.     

Simultaneous decomplexification of a pair of complex matrices by a similarity transformation

The following question is treated: Under what conditions can complex n-by-n matrices A and B be made real by the same similarity transformation? It is shown that if the algebra generated by A and B contains a matrix with a simple real spectrum, then the problem of the simultaneous decomplexification of a matrix pair can be reduced to the decomplexification of a single matrix by a diagonal similarity transformation. From this result, sufficient conditions are derived for the possibility of simultaneous decomplexification. An example illustrating these conditions is given.     

Eigenvalue inequalities for principal submatrices

For a polynomial with real roots, inequalities between those roots and the roots of the derivative are demonstrated and translated into eigenvalue inequalities for a hermitian matrix and its submatrices. For example, given an n-by-n positive definite hermitian matrix with maximum eigenvalue λ, these inequalities imply that some principal submatrix has an eigenvalue exceeding $\text{[(}\text{n?1)}\text{n}\text{]λ}$.     

Matrix multiplication is determined by orthogonality and trace

Any associative bilinear multiplication on the set of n-by-n matrices over some field of characteristic not two, that makes the same vectors orthogonal and has the same trace as ordinary matrix multiplication, must be ordinary matrix multiplication or its opposite.     

Schur-type stability properties and complex diagonal scaling

We introduce new classes of n-by-n matrices with complex entries which can be scaled by a diagonal matrix with complex entries to be normal or Hermitian and study the Schur-type stability properties of these matrices.     

The inverse eigenvalue problem of reflexive matrices with a submatrix constraint and its approximation

In this paper, we first consider the existence of and the general expression for the solution to the constrained inverse eigenvalue problem defined as follows: given a generalized reflection matrix P∈R ^n×n , a set of complex n-vectors {x _i } _i=1 ^m , a set of complex numbers {λ _i } _i=1 ^m , and an s-by-s real matrix C ₀, find an n-by-n real reflexive matrix C such that the s-by-s leading principal submatrix of C is C ₀, and {x _i } _i=1 ^m and {λ _i } _i=1 ^m are the eigenvectors and eigenvalues of C, respectively. We are then concerned with the best approximation problem for the constrained inverse problem whose solution set is nonempty. That is, given an arbitrary real n-by-n matrix $\tilde{C}$ , find a matrix C which is the solution to the constrained inverse problem such that the distance between C and $\tilde{C}$ is minimized in the Frobenius norm. We give an explicit solution and a numerical algorithm to the best approximation problem. An illustrative experiment is also presented.     

Lattice-Ordered Matrix Rings Over Totally Ordered Rings

For an n ×n matrix algebra over a totally ordered integral domain, necessary and sufficient conditions are derived such that the entrywise lattice order on it is the only lattice order (up to an isomorphism) to make it into a lattice-ordered algebra in which the identity matrix is positive. The conditions are then applied to particular integral domains. In the second part of the paper we consider n ×n matrix rings containing a positive n-cycle over totally ordered rings. Finally a characterization of lattice-ordered matrix ring with the entrywise lattice order is given.     

A Characterization of Positive Matrices

It is shown that an n-by-n matrix has a strictly dominant positive eigenvalue with positive left and right eigenvectors and this property is inherited by principle submatrices if and only if it is entry-wise positive. This limits the extent to which attractive Perron-Frobenius properities may be generalized outside the positive matrices.Mathematics Subject Classification (2000): 15A48     

The moore-penrose inverse of a matrix over a semi-simpie artinian ring

Given an m-by-n matrix over a semi-simple artinian ring with aninvolutory automorphism, necessary and sufficient conditions are given for the matrix tao have a Moore—Penrose inverse. Moreover, expressions for the MP-inverse are obtained. These formulas may be considered as a generalization of the MacDuffee formula for the MP-inverse of a matrix over the complex numbers.     

The extent to which triangular sub-patterns explain minimum rank

For any zero-nonzero pattern of a matrix, the minimum possible rank is at least the size of a sub-pattern that is permutation equivalent to a triangular pattern with nonzero diagonal. For certain numbers of rows and columns, the minimum rank of a pattern is k only when there is a k-by-k such triangle. Here, we complete the determination of such sizes by showing that an m-by-n pattern of minimum rank k must contain a k-triangle for m=5, k=4; m=6, k=5; and m=6, k=4. A table is given showing whether or not this happens for all m, n, k. In the process, a Schur complement approach to minimum rank is described and used, and simple ways to recognize the presence of triangles of sizes less than 7 are given.     

The endomorphism spectrum of a monounary algebra

The endomorphism spectrum specA of an algebra A is defined as the set of all positive integers, which are equal to the number of elements in an endomorphic image of A, for all endomorphisms of A. In this paper we study finite monounary algebras and their endomorphism spectrum. If a finite set S of positive integers is given, one can look for a monounary algebra A with S = specA. We show that for countably many finite sets S, no such A exists. For some sets S, an appropriate A with spec A = S are described. For n ∈ ? it is easy to find a monounary algebra A with {1, 2, ..., n} = specA. It will be proved that if i ∈ ?, then there exists a monounary algebra A such that specA skips i consecutive (consecutive eleven, consecutive odd, respectively) numbers. Finally, for some types of finite monounary algebras (binary and at least binary trees) A, their spectrum is shown to be complete.     

Nonsingularity of least common multiple matrices on gcd-closed sets

Let n be a positive integer. Let S={x₁,…,x_n} be a set of n distinct positive integers. The least common multiple (LCM) matrix on S, denoted by [S], is defined to be the n×n matrix whose (i,j)-entry is the least common multiple [x_i,x_j] of x_i and x_j. The set S is said to be gcd-closed if for any x_i,x_j∈S,(x_i,x_j)∈S. For an integer m>1, let ω(m) denote the number of distinct prime factors of m. Define ω(1)=0. In 1997, Qi Sun conjectured that if S is a gcd-closed set satisfying max_x∈S{ω(x)}?2, then the LCM matrix [S] is nonsingular. In this paper, we settle completely Sun's conjecture. We show the following result: (i). If S is a gcd-closed set satisfying max_x∈S{ω(x)}?2, then the LCM matrix [S] is nonsingular. Namely, Sun's conjecture is true; (ii). For each integer r?3, there exists a gcd-closed set S satisfying max_x∈S{ω(x)}=r, such that the LCM matrix [S] is singular.     

On the total positivity of restricted Stirling numbers

This note shows that the matrix whose (n,k) entry is the number of set partitions of {1,…,n} into k blocks with size at most m is never totally positive for m≥3; thus answering a question posed in [H. Han, S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, European J. Combin. 29 (2008) 1544-1554].