📂Matrix Algebra

Matrix Definitions

Definition¹

A matrix is an arrangement of numbers in the shape of a rectangle as follows:

$$ A=\begin{bmatrix} 10 & 0 & 3 \\ 0 & 8 & 22 \end{bmatrix} $$

Each of the arranged numbers is called an entry or element. A horizontal line is called a row, and a vertical line is called a column. Moreover, if a certain matrix has $m$ rows and $n$ columns, its size is denoted by $m \times n$.

In the example above, the matrix $A$ has 2 rows and 3 columns, and its size is $2\times 3$. It is important to note that $\times$ does not mean multiplication. The size must be indicated clearly with the number of rows and columns, as in $2\times 3$, and should never be written as $6$. For reference, a '$2 \times 3$ matrix' is read as [two by three matrix].

Notation

Matrices are usually denoted as below with either square brackets [] or parentheses (), both of which are commonly seen. However, when writing by hand, it can be difficult to make parentheses look nice. Unlike denoting 2D or 3D space coordinates, it is standard to not place commas (,) between elements.

$$ A=\begin{bmatrix} 10 & 0 & 3 \\ 0 & 8 & 22 \end{bmatrix} \quad A=\begin{pmatrix} 10 & 0 & 3 \\ 0 & 8 & 22 \end{pmatrix} $$

Typically, matrices are denoted in uppercase, and their elements in lowercase. For example, the element in the 1st row and 3rd column of matrix $A$ is $3$, and this is denoted as follows.

$$ a_{13}=3 $$

The first subscript indicates the row position, and the second subscript indicates the column position. Similarly, the element in the $i$th row, $j$th column of a matrix, if it is $a_{ij}$, is denoted as $\begin{bmatrix} a_{ij} \end{bmatrix}$. The $(i,j)$ element of $A$ is denoted as $[A]_{ij}$.

$$ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} = \begin{bmatrix} a_{ij} \end{bmatrix},\qquad [A]_{ij} = a_{ij} $$

The set of all $m\times n$ matrices is denoted as follows.

$$ M_{m \times n} $$

The set of matrices of size $m\times n$ with real $\mathbb{R}$, complex $\mathbb{C}$ elements are denoted respectively as follows.

$$ M_{m\times n}(\mathbb{R}),\quad M_{m \times n}(\mathbb{C}) $$

More abstractly, the set of $n \times n$ matrices with elements in the field $F$ are denoted as $M_{m \times n}(F)$.

Column Vectors and Row Vectors

A vector is an arrangement of numbers horizontally or vertically. Thinking about this, a matrix can be seen as an arrangement of column vectors or row vectors. Let’s look at the matrix $A$ that has been continuously used as an example.

$$ A= \begin{bmatrix} 10 & 0 & 3 \\ 0 & 8 & 22 \end{bmatrix} $$

Each column of $A$ can be thought of as comprising the column vectors $\begin{bmatrix} 10 \\ 0 \end{bmatrix}$, $\begin{bmatrix} 0 \\ 8 \end{bmatrix}$, $\begin{bmatrix} 3 \\ 22 \end{bmatrix}$. Alternatively, it can be seen that each row consists of the row vectors $\begin{bmatrix} 10 & 0 & 3 \end{bmatrix}$, $\begin{bmatrix} 0 & 8 & 22 \end{bmatrix}$.