This paper describes a generalization of the inverse of a non-singular matrix, as the unique solution of a certain set of equations. This generalized inverse exists for any (possibly rectangular) matrix whatsoever with complex elements. It is used here for solving linear matrix equations, and among other applications for finding an expression for the principal idempotent elements of a matrix. Also a new type of spectral decomposition is given.

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A Generalized inverse for matrices

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

(1958). Pseudo-Inverses in Associative Rings and Semigroups. The American Mathematical Monthly: Vol. 65, No. 7, pp. 506-514.

Pseudo-Inverses in Associative Rings and Semigroups

Let S be any semigroup, i.e. a set furnished with an associative binary operation, denoted by juxtaposition. An involution on S will mean a bijection u —• u* of S onto itself, satisfying (a*)* = a, (ab)* = b*a*. Such an involution is called proper (cf. [3, p. 74] ) iff a*a = a*b = b*a = b*b implies a = b. A proper ̂ semigroup will mean a pair (5, *) where 5 is a semigroup and * is a specified proper involution of S; in practice, we write S as an abbreviation for (S, *). From now on, 5 will denote an arbitrary proper *-semigroup. Obvious natural special cases are (i) all proper *-rings, with \"properness\" (Herstein [4, p. 794] prefers to say \"positive deflniteness\") as customarily defined (cf. [5, p. 31], [1, p. 10]) via u*u = 0 implying u = 0 (in particular, with the obvious choices for *, all commutative rings with no nonzero nilpotent elements, all Boolean rings, the ring B(S) of all bounded linear operators on any complex Hubert space H, and the ring Mn(C) of all n x n complex matrices), and, only slightly less trivially, (ii) all inverse semigroups (in particular, all groups). We make a start here towards showing that the proper *-semigroup axioms allow the simultaneous development of a surprisingly rich common theory of these special cases. While there is clearly little likelihood of learning anything new about groups or Boolean rings by such an approach, none of the results about proper *-semigroups which we state below has previously been noted even for n x n matrices (still less for B(H) or for *-rings), and most provide new information also about inverse semigroups. We recall that an element a E S is called regular iff a E aSa, and *-regular iff there is an x E S with

https://projecteuclid.org/download/pdf_1/euclid.bams/1183540393

Natural structures on semigroups with involution

A class of outer generalized inverses

Some Theorems on Commutative Matrices

For any semigroup and any , in 2012, the author defined to be a -inverse of if , and . For given , any such is unique if it exists, and, by choosing and appropriately, one can arrange for to become, inter alia, either the author’s pseudo-inverse or (if is a -semigroup) the Moore–Penrose inverse . For any , the author proved in 1958 that implies whenever exists. In this article it is shown that, more generally, implies whenever and both exist. Still more generally, a corresponding result is proved for -inverses; in particular, for the Moore–Penrose inverse, and together imply . For any , new connections between the generalized invertibility of and are also obtained.

Michael P. Drazin

Papers

Pseudo-Inverses in Associative Rings and Semigroups

Natural structures on semigroups with involution

A class of outer generalized inverses

Some Theorems on Commutative Matrices

Commuting properties of generalized inverses