保体上上三角幂等矩阵的诱导映射

假设T_m(D)是体D上所有上三角m×m矩阵的集合.首先分别给出诱导映射和保幂等性的定义.然后为了刻画T_m(D)的保幂等的诱导映射,提出类序列的概念,同时描述类序列的性质.最后,使用矩阵技术和初等方法,借助于分类讨论得到了T_m(D)的保幂等的诱导映射的一般形式并且给出了某些例子,用以解释某些结果之间的关系.     

On the trace of idempotent matrices over group algebras

Let G be a group and E an idempotent matrix with entries in the group algebra CG. In this paper, we consider the embedding of CG into the von Neumann algebra G and use the center-valued trace on the latter, in order to obtain some information about the coefficients of the Hattori-Stallings rank of E. Our results generalize the inequalities obtained previously by Kaplansky [11], Passi, Passmann, Luthar and Alexander [1,10,12], while providing at the same time a unified and coherent presentation of these, via the notion of moments that are associated with E.     

Perimeter preserver of matrices over semifields

For a rank-1 matrix A = ab ^t, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over semifields. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices over semifields if and only if it has the form T(A) = U AV, or T(A) = U A ^t V with some invertible matrices U and V. This work was supported by the research grant of the Cheju National University in 2006.     

Products of idempotent matrices over integral domains

We say that a ring R has the idempotent matrices property if every square singular matrix over R is a product of idempotent matrices. It is known that every field, and more generally, every Euclidean domain has the idempotent matrices property. In this paper we show that not every integral domain has the idempotent matrices property and that if a projective free ring has the idempotent matrices property then it must be a Bezout domain. We also show that a principal ideal domain has the idempotent matrices property if and only if every fraction a/b with b≠0 has a finite continued fraction expansion. New proofs are also provided for the results that every field and every Euclidean domain have the idempotent matrices property.     

Products of idempotent matrices over Hermite domains

Communicated by F. Pastijn     

On sums of idempotent matrices over a field of positive characteristic

We study which square matrices are sums of idempotents over a field of positive characteristic; in particular, we prove that any such matrix, provided it is large enough, is actually a sum of five idempotents, and even of four when the field is a prime one.     

On linear combinations of two idempotent matrices over an arbitrary field

Given an arbitrary field K and non-zero scalars α and β, we give necessary and sufficient conditions for a matrix A∈M_n(K) to be a linear combination of two idempotents with coefficients α and β. This extends results previously obtained by Hartwig and Putcha in two ways: the field K considered here is arbitrary (possibly of characteristic 2), and the case α≠±β is taken into account.     

On the Kapranov ranks of tropical matrices

We investigate the Kapranov rank functions of tropical matrices for different ground fields. For any infinite ground field we show that the rank-product inequality holds for Kapranov rank, and we prove that the Kapranov rank respects Green’s preorders on the semigroup of tropical n-by-n matrices. The rank-product inequality is shown to fail for Kapranov rank over any finite ground field. We provide an example of a 7-by-7 01-matrix whose Kapranov rank is independent of a ground field, equals 6, and exceeds tropical rank.     

On the structure of semigroups of idempotent matrices

We prove that any pure regular band of matrices admits a simultaneous LU decomposition in the standard form. In case that such a band forms a double band called a skew lattice, we obtain the standard form without the assumption of purity.     

Linear preservers of term ranks of matrices over semirings

The term rank of a matrix $A$ over a semiring $\mathbf{S}$ is the least number of lines (rows or columns) needed to include all the nonzero entries in $A$. In this paper, we study linear operators that preserve term ranks of matrices over $\mathbf{S}$. In particular, we show that a linear operator $T$ on matrix space over $\mathbf{S}$ preserves term rank if and only if $T$ preserves term ranks $1$ and $\alpha (\ge 2)$ if and only if $T$ preserves two consecutive term ranks in a restricted condition. Other characterizations of term-rank preservers are also given.     

Triangularizing semigroups of matrices over a skew field

We define a closure operation on semigroups of matrices over a skew field, and show that a semigroup of matrices can be (upper) triangularized if and only if its closure can be. We then give necessary and sufficient conditions for a closed semigroup to be triangularizable.     

On the ranks of principal submatrices of diagonalizable matrices

As is well known, the rank of a diagonalizable complex matrix can be characterized as the maximum order of the nonzero principal minors of this matrix. The standard proof of this fact is based on representing the coefficients of the characteristic polynomial as the (alternating) sums of all the principal minors of appropriate order. We show that in the case of normal matrices, one can give a simple direct proof, not relying on those representations. Bibliography: 2 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 42–44.     

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.     

A note on idempotent matrices

Let H be an n × n matrix, and let the trace, the rank, the conjugate transpose, the Moore-Penrose inverse, and a g-inverse (or an inner inverse) of H be respectively denoted by trH, ρ(H), H¹, H², and H^?. This note develops two results: (i) the class of idempotent g-inverse of an idempotent matrix, and (ii) if H is an n × n matrix and ρ(H) = trH, then $\text{tr}\text{(H}{}^2\text{H}{}^2\text{H}{}^1\text{) ? ρ(H)}$, and the equality holds iff H is idempotent. This result is compared with the previous result of Khatri (1983), and some consequences of (i) and (ii) are given.