Real Matrices
Why
We compress the notation for linear equations.
Definition
A real matrix
(matrix,
rectangular array) is a
two-dimensional array of real numbers.
We denote the elements of a matrix in a grid
between two rectangular braces, as in
\[
A = \bmat{
A_{11} & A_{12} & \cdots & A_{1n} \\
A_{21} & A_{22} & \cdots & A_{2n} \\
\vdots & & \ddots & \vdots \\
A_{m1} & A_{m2} & \cdots & A_{mn} \\
}.
\]
We call $m$ and $n$ the dimensions of the matrix. We call $m$ the height and $n$ the width. If the height of the matrix is the same as the width of the matrix then we call the matrix square. If the height is larger than the width, we call the matrix tall. If the width is larger than the height, we call the matrix wide.
Linear equations
Recall that we are interested in solutions of
the linear equations
\[
\begin{aligned}
y_1 &= A_{11}x_1 + A_{12}x_2 + \cdots + A_{1n}x_n, \\
y_2 &= A_{21}x_1 + A_{22}x_2 + \cdots + A_{2n}x_n, \\
&\vdots \\
y_n &= A_{m1}x_1 + A_{m2}x_2 + \cdots + A_{mn}x_n. \\
\end{aligned}
\]
A primer on matrix-vector products
Using the notation $A \in \R ^{m \times n}$
and $x \in \R ^n$ we want a compressed way to
write the above system of linear equations.
Define the real matrix-vector
product $z \in \R ^m$ of $A$ with $x$ by
\[
z_{i} = \sum_{j = 1}^{n} A_{ij}x_j, \quad i = 1, \dots , m.
\]
Notation
We express the above system of linear equations
as
\[
y = Ax,
\]
Note on terminology
The word “matrix” is from the Latin “mater,” meaning mother, and has an old sense in English similar “womb.” The matrix is source of many determinants (discussed elsewhere).