Separation Transformations for Square Matrices

A class of transformations of square matrices which may be regarded as a generalization of matrix transposition and complex conjugation is defined by a set of five postulates. These transformations, termed "separation transformations," make evident certain characteristics of form for square matrices by resolving every square matrix into two additive component matrices called the "consonant" and "dissonant" parts. This property may be valuable in the study of active linear networks since the impedance matrices, for example, of these networks may display no obvious special characteristic of form. Separation transformations and the related consonant and dissonant parts of square matrices have some interesting manipulative properties. These properties lead to an extension of the concept of orthogonality of matrices. An example to illustrate the possible use of the relations derived for separation transformations shows a generalization of Mason\´s invariant for active linear networks subjected to lossless, linear, reciprocal coupling.