The notion of the inverse of a matrix with entries from the real or complex fields was generalized by Moore (6, 7) in 1920 to include all rectangular (finite dimensional) matrices. In 1951, Bjerhammar (2, 3) rediscovered the generalized inverse for rectangular matrices of maximal rank. In 1955, Penrose (8, 9) independently rediscovered the generalized inverse for arbitrary real or complex rectangular matrices. Recently, Arghiriade (1) has given a set of necessary and sufficient conditions that a matrix commute with its generalized inverse. These conditions involve the existence of certain submatrices and can be expressed using the notion of EPr matrices introduced in 1950 by Schwerdtfeger (10). The main purpose of this paper is to prove the following theorem:Theorem 2. A necessary and sufficient condition that the generalized inverse of the matrix A (denoted by A+) commute with A is that A+ can be expressed as a polynomial in A with scalar coefficients.

No abstract

This pape r presents three s imple mathematical mode ls, all of game-theo retic type, d ea lin g with an inspector-inspectee relationship. The in s pectee alw ays tries to m ax imize hi s ne t gain, whic h is the amount he obtain s by "cheatin g" less the amount he is penalize d wh e n ca ught. Th e first mode l assumes a zero-sum payoff and so the in spector tries to minimize the in spectee's ne t ga in . [n th e second mod e l, the in s pec tor tries to deter c heat in g wit hout co nce rn fo r the ex tract ion of pe nalties.[n the third mode l we assume that th e probabilistic patte rn of the in s pecto r' s stra tegy is kn ow n to the in s pectee and that the inspector co nstructs hi s strat egy with thi s in mind. Each of these mode ls is analyzed and op timal so lution s are obtain ed . Seve ral s imple exa mples are prese nt e d to show th e re lat ion betw ee n the leve l of c hea tin g and the levels of in s pec tio n reso urces and penalty.

(1) EPr matrices A (that is, matrices A tor which A and A * hav e the same null space) are investi· gated. It is shown that if A is a complex EPr, matrix and B a com pl ex EPr2 matrix, and AB = BA, then AB is EPr. Other theorems about products of EPr matrices are es tablished.(2) Let A be a normal EP,. matrix over an arbitrary field . A necessa ry and sufficient condition, involving the solvability (for X) of a matrix equationXBX*+AX + X*A*+ C = O, is found for the exist· ence of a matrix N s uch that (i) NN*=/ and (ii) A*=NA =AN. Explicit solutions are given for two important classes of normal EPr matrices, namely (1) those sati sfying the condition rank A = rank AA *, and (2) those of rank n/2, satis fying AA * = 0, over a fi eld of c harac teristic ¥ 2. An example is given to s how that no suc h N need ex is t for c haracteris ti c =2.(3) EP lin ear tran sfor mation s on a finite·dimensiona l vector s pace a re introdu ced, and the re lation between th e m and EPr matrices is s tudi ed. It is shown that a lin ea r transformation T of a comp le x vector space is EP if and on ly if rank T = rank P.

No abstract