Arnold Richard Kräuter

On some questions concerning permanents of (1,−1)-matrices

Let Ωn denote the set of alln×n-(1,−1)-matrices. E.T.H. Wang has posed the following problem: For eachn≧4, can one always find nonsingularA∈Ωn such that |perA|=|detA| (*)? We present a solution forn≦6 and, more generally, we show that (*) does not hold ifn=2k−1,k≧2, even for singularA∈Ωn. Moreover, we prove that perA≠0 ifA∈Ωn,n=2k−1, and we derive new results concerning the divisibility of the permanent in Ωn by powers of 2.

Lower bounds for the determinant and the trace of a class of Hermitian matrices

Abstract Sharp lower bounds for the determinant and the trace of a certain class of hermitian matrices are derived. Special attention is given to the discussion of the case of equality in these estimations. Since the trace turns out to be a special kind of condition number, our results are helpful tools in selecting measurement directions needed in several physical problems. This application is also discussed in detail.

An upper bound for the permanent of a nonnegative matrix

Abstract Let A be a fully indecomposable, nonnegative matrix of order n with row sums r l , r n , and let s i equal the smallest positive element in row i of A . We prove the permanental inequality per (A)⩽ ∏ i=1 n s i + ∏ i=1 n (r i −s i ) and characterize the case of equality. In 1984 Donald, Elwin, Hager, and Salamon gave a graph-theoretic proof of the special case in which A is a nonnegative integer matrix.

Some properties of the permanent of (1, -1)-matrices

Let ω n denote the set of all n × n −(1, −1)-matrices. In [5] E. I. II. Wang posed the following problem. Is there a decent upper bound for [per A] when A ∊ ω n is nonsingular? In this paper we conjecture that the best possible bound is the permanent of a matrix in ω n which contains exactly n 1 negative entries all occurring in its main diagonal. This conjecture is affirmed by the study of a large class of matrices in Ω n . Moreover, some other interesting results concerning the permanent function in Ω n are given.

On convertible (0,1)-matrices

A matrix A = (aij ) will be called convertible if there exists a matrix S(A) = (±aij ) such that per A = detS(A). P. M. Gibson [3] proved that a convertible n × n-(0, l)-matrix with positive permanent contains at least zeros. In this paper we consider the class of those matrices A with exactly Ω n zeros, and we determine all possible numbers of changes of signs induced by the mapping S. Furthermore, for the least and largest of these numbers the matrices S(A) are completely characterized.

On the greatest distance between two permanental roots of a matrix

Abstract The permanental spread of a complex square matrix A is defined to be the greatest distance between two roots of the equation per(zI − A) = 0. A preliminary study of this number as well as of two related quantities is given. In particular, we derive upper and lower bounds and deal with comparisons of different bounds. Finally, two inequalities involving the permanental spread are treated.

On a theorem of M. Marcus and H. Mine and a little known expansion formula for permanents

Trying to reveal the actual origin of a theorem on the permanent of the sum of two matrices, we discovered a little known formula concerning the expansion of determinants by determinants of bordered matrices. In this note, we present a concise and self-contained proof of the permanental analogue of this formula. Finally, a connection with the beforementioned theorem is established and applications of either result are referred to.

Comparison of permanental bounds of (0,1)-matrices

Abstract Let A be a nonnegative integral n -square matrix with row sums r 1 , …, r n . It is known that per A ⩽ Π n i=1 r i ! l r i if A is a (0, 1)-matrix (Minc, 1963; Bregman, 1973) and also that per A ⩽ 1 + Π n i = 1 ( r i − 1) if A is fully indecomposable (Donald et al., 1984). These two bounds are, in general, uncomparable, even in the case that A is a fully indecomposable (0, 1)-matrix. In this paper we obtain some comparison test for these bounds with the aid of a function involving the gamma function.