ABSTRACT

This chapter describes the basics of matrix arithmetic and matrix algebra and matrix theory proper with an introduction to diagonalization, an idea of fundamental importance in mathematics. The array of numbers that comprises a matrix is enclosed in round brackets although in some books square brackets are used. Most of the matrices have either real or complex entries, but there is no reason why matrices should not have entries from more exotic kinds of numbers such as the modular arithmetic systems. The fact that matrix multiplication is not commutative is one of the reasons that quantum mechanics is so different from classical mechanics. The theory of quantum computing makes heavy use of Hermitian matrices and their properties. Systems of linear equations in several variables cannot apparently be solved in such a straightforward way. But by using matrices, the system can be packaged into a single matrix equation in one vector variable.