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(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.

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