Relative EP matrices

The purpose of the present work is to introduce the concept of relative EP matrix of a rectangular matrix relative to a partial isometry (or, in short, $T$-EP matrix) hitherto unknown. We extend various basic results on EP matrices and we study the relationship between $T$-hermitian, $T$-normal and $T$-EP matrices. The main theorems of this paper consist in providing canonical forms of relative EP matrices when matrices involved are rectangular as well as square. We then use them to characterize the relative EP matrices and show their properties. In fact, an interesting fact that has emerged is that $A$ is $T$-EP if and only if there is an EP matrix $C$ such that $A=CT$ and $C=TT^*C$ whatever be the matrix, square or rectangular. We also give various necessary and sufficient conditions for a matrix to be $T$-EP.