Abstract
This chapter discusses the elements of matrix calculus. In a bra or row-vector <x| components are ordered horizontally, and in a ket or column-vector |y> components are ordered vertically. The components of the row of the first vector, bra or ket, are multiplied by the corresponding components of the column of the second vector, ket or bra, and if there are several results they are added. The interaction of two vectors can, depending on the order of operations, generate two different mathematical objects, that is, either numbers or matrices. The inner and outer products bridge two important mathematical domains—the domain of numbers and the domain of matrices. The outer product serves to define the closure relation that can be employed to construct a vector space. The inner product offers a way of defining the orthogonality of vectors. The complete vector space can be generated from vectors called basis vectors and the key property of a basis set of vectors is linear independence.
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