# Unique builders for classes of matrices

## TL;DR

This paper introduces unique basic matrices for normal matrices, enabling new representations and algorithms for quantum gates, idempotents, and pseudo-inverses, with broad applications in matrix theory and quantum computing.

## Contribution

It develops a framework of unique basic matrices for normal matrices and derives algorithms and formulas for quantum gates, idempotents, and pseudo-inverses.

## Key findings

- Unique basic matrices for normal matrices are established.
- An efficient algorithm for expressing idempotents as sums of rank 1 idempotents is presented.
- Formulas for symmetric unitary matrices and pseudo-inverses are derived.

## Abstract

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.