Unique builders for classes of matrices

Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived since a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank $1$ idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) $U$ is a symmetric unitary matrix if and only if it has the form $I-2E$ for a symmetric idempotent $E$, (ii) a formula for the pseudo inverse in terms of basic matrices is derived. Examples for various uses are readily available.