Abstract: A number of problems from specifiers for Boolean networks to programs for quantum computers can be encoded as matrices. The paper presents a novel family of linear, generative representations for evolving matrices. The matrices can be general or restricted within special classes of matrices like permutation matrices, Hermitian matrices, or other groups of matrices with particular algebraic properties. These classes include unitary matrices which encode quantum programs. This representation avoids the brittleness that arises in direct representations of matrices and permits the researcher substantial control of the part of matrix space being searched. The representation is demonstrated on a relatively simple matrix problem in automatic content generation as well as Boolean map induction and automatic quantum programming. The automatic content generation problem yields interesting results; the generative matrix representation yields worse fitness but a substantially greater variety of outcomes than a direct encoding, which is acceptable when generating content. The Boolean map experiments extend and confirm results that demonstrate that the generative encoding is superior to a direct encoding for the transition matrix of a Boolean map. The quantum programming results are generally quite good, with poor performance on the simplest problems in two of the families of programming tasks studied. The viability of the new representation for evolutionary matrix induction is well supported.