Introductory Linear Algebra/Matrices

Motivation

One important application for matrices is solving systems of linear equations. Some of the following definitions may be viewed as 'designed for solving system of linear equations'.

Some terminologies

An $m\times n$ (read 'm by n') is a matrix with $m$ rows and $n$ columns, and $m\times n$ is the Template:Colored em of the matrix. The rows are counted from the top, and the columns are counted from the left. If the size of a matrix is $1\times 1$, we simply refer to this matrix as a Template:Colored em, and no brackets are needed in this case. The Template:Colored em of all $m\times n$ matrices with Template:Colored em is denoted by $M_{m\times n}(ℝ)$. A Template:Colored em letter is usually used to denote a Template:Colored em, while Template:Colored em letters are used to denote its Template:Colored em. For example, $A=(a_{ij}{)}_{m\times n}$ denotes an $m\times n$ matrix $A$ with entries $a_{ij}$ in which $1\le i\le \underset{\text{no. of rows}}{\underbrace{m}}$ and $1\le j\le \underset{\text{no. of columns}}{\underbrace{n}}$. (We may omit the subscript specifying the size of matrix if its size is already mentioned, or its size is not important.) Template:Colored definition Template:Colored remark Template:Colored exercise Template:Colored exercise Template:Hide In particular, if a matrix has the same number of rows and columns, then it has some nice properties. In view of the shape of such a matrix (square-like), we define such matrices as Template:Colored em. Template:Colored definition We will also introduce a term, namely Template:Colored em, which will be useful in some situations. Template:Colored definition Template:Colored example Template:Colored remark Then, we will define some types of matrices for which the definitions are related to the Template:Colored em. Template:Colored definition

Template:Colored remark Template:Colored definition Template:Colored remark Template:Colored exercise The last terminology we mention here is Template:Colored em, which will sometimes be used. Template:Colored definition Template:Colored remark Template:Colored exercise

Matrix operations

In this section, we will cover different matrix operations. Some operations are quite different from that in the number system, in particular, matrix multiplication. Template:Colored definition Template:Colored definition Then, we are going to define Template:Colored em, which is quite different from the multiplication in the number system. Template:Colored definition On the other hand, a positive Template:Colored em of a Template:Colored em is defined quite similarly to that in number system. Template:Colored definition

Then, we will discuss matrix analogs for the numbers zero and one in the number system, namely the Template:Colored em and the Template:Colored em, which, in the number system, are analogous to the numbers $0$ and $1$ respectively. Template:Colored definition Template:Colored remark Template:Colored definition Template:Colored remark Template:Colored example Template:Colored proposition Template:Colored remark Then, we will introduce an operation that does not exist in the number system, namely transpose. Template:Colored definition

Proof. It follows from observing that matrix transpose has the same size as the original matrix Template:Colored em the matrix is Template:Colored em, since the number of rows and number of columns are swapped for the matrix transpose, and the size remains unchanged if and only if the number of rows equals number of columns.

$◻$