Matrices

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A matrix is a rectangular array whose entries are of the same type.

In the previous chapter we have already invoked the concept of matrices. However before going further we will need to establish a framework for them, and some of their basic properties.

We begin by recalling some notation and terminology. A matrix will mean a rectangular array whose entries are of the same type. Thus, we could have an array of real numbers, complex numbers, functions, or even matrices. An $m\times n$ matrix will refer to one which has $m$ rows and $n$ columns, and the collection of all $m\times n$ matrices of real numbers will be denoted by $\mathbb R^{m\times n}$ . We adopt the convention, used by MATLAB, in which the $(i,j)^{th}\ entry$ of the matrix $A$ (that in row $i$ and column $j$ ) is denoted by $A(i,j)$ . Also, following MATLAB notation, we will write the $i^{th}$ row as $A(i,:)$ , and the $j^{th}$ column as $A(:,j)$ . Before getting to the operations themselves, we first record

Two matrices $A$ and $B$ are equal if $A(i,j) = B(i,j)$ for all $i,j$ .

Note that this equality forces $A$ and $B$ to have the same dimensions, because if they had different dimensions, there would have to be a choice of indices $(i,j)$ for which one side of the equation exists, but the other does not. Thus, equality can be reformulated as saying: $A$ and $B$ have the same dimensions, and the same entry in each place.

A matrix $A$ is an $m\times n$ matrix if it has $m$ rows and $n$ columns. The numbers $m$ and $n$ are referred to as the dimensions of $A$ .

In this course we will only be concerned with finite matrices; those with finitely many rows and columns. Observe that in order to precisely specify what a given matrix $A$ is, it suffices to know i) its dimensions $m$ and $n$ and ii) the value of the entry $A(i,j)$ for each $1\le i\le m$ and $1\le j\le n$ .

When working with a fixed system of numbers such as $\mathbb R$ (real numbers) or $\mathbb C$ (complex numbers), the elements of that number system in linear algebra are often referred to as scalars.