Matrices

Abstract

A rectangular array

$$A = \left( {\begin{array}{*{20}{c}} {\alpha _{1}^{1}} & \cdots & {\alpha _{1}^{m}} \\ \vdots & {} & \vdots \\ {\alpha _{n}^{1}} & \cdots & {\alpha _{n}^{m}} \\ \end{array} } \right)$$

((3.1))

) of n m scalars $\alpha _{v}^{\mu }$ is called a matrix of n rows and m columns or, in brief, an n×m-matrix. The scalars $\alpha _{v}^{\mu }$ are called the entries or the elements of the matrix A. The rows

$${a_{v}} = (\alpha _{v}^{1} \ldots \alpha _{v}^{m})\quad (v = 1 \ldots n)$$

can be considered as vectors of the space Γ^m and therefore are called the row-vectors of A. Similarly, the columns

$${b^{\mu }} = (b_{1}^{\mu } \ldots b_{n}^{\mu })\quad (\mu = 1 \ldots m)$$

considered as vectors of the space Γⁿ, are called the column-vectors of A.

In this chapter all vector spaces will be defined over a fixed, but arbitrarily chosen field Γ of characteristic 0.