Matrix (mathematics)

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In mathematics, a matrix (plural matrices) is a rectangular table of numbers. There are rules for adding, subtracting and "multiplying" matrices together. These rules sometimes lead to not so common properties, for example, if A and B are two matrices, it is not always true that $A \cdot B$ equals $B \cdot A$ .

Many natural sciences use matrices quite a lot. In many universities, courses about matrices (usually called linear algebra) are taught very early, sometimes even in the first year of studies.

1 Definitions and notations
- 1.1 Example
2 Operations
- 2.1 Addition
- 2.2 Multiplication of two matrices
3 Special matrices
4 One column matrix

[edit] Definitions and notations

The horizontal lines in a matrix are called rows and the vertical lines are called columns. A matrix with m rows and n columns is called an m-by-n matrix (or m×n matrix) and m and n are called its dimensions.

The places in the matrix where the numbers are, are called entry. The entry of a matrix A that lies in the row number i and column number j is called the i,j entry of A. This is written as A[i,j] or a_ij.

We write $A:=(a_{ij})_{m \times n}$ to define an m × n matrix A with each entry in the matrix called a_ij for all 1 ≤ i ≤ m and 1 ≤ j ≤ n.

[edit] Example

The matrix

$\begin{bmatrix} 1 & 2 & 3 \\ 1 & 2 & 7 \\ 4 & 9 & 2 \\ 6 & 1 & 5 \end{bmatrix}$

is a 4×3 matrix. This matrix has m=4 rows, and n=3 columns.

The element A[2,3] or a₂₃ is 7.

[edit] Operations

[edit] Addition

The sum of two matrices is the matrix, which (i,j)-th entry is equal to the sum of the (i,j)-th entries of two matrices:

$\begin{bmatrix} 1 & 3 & 2 \\ 1 & 0 & 0 \\ 1 & 2 & 2 \end{bmatrix} + \begin{bmatrix} 0 & 0 & 5 \\ 7 & 5 & 0 \\ 2 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 1+0 & 3+0 & 2+5 \\ 1+7 & 0+5 & 0+0 \\ 1+2 & 2+1 & 2+1 \end{bmatrix} = \begin{bmatrix} 1 & 3 & 7 \\ 8 & 5 & 0 \\ 3 & 3 & 3 \end{bmatrix}$

The two matrices have the same dimensions. Here $A + B = B + A$ is true.

[edit] Multiplication of two matrices

The multiplication of two matrices is a bit more elaborated:

$\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ \end{bmatrix} \cdot \begin{bmatrix} 6 & -1 \\ 3 & 2 \\ 0 & -3 \end{bmatrix} = \begin{bmatrix} (1 \cdot 6 + 2 \cdot 3 + 3 \cdot 0) & (1 \cdot -1 + 2 \cdot 2 + 3 \cdot -3) \\ (4 \cdot 6 + 5 \cdot 3 + 6 \cdot 0) & (4 \cdot -1 + 5 \cdot 2 + 6 \cdot -3) \\ \end{bmatrix} = \begin{bmatrix} 12 & -6 \\ 39 & -12 \end{bmatrix}$

two matrices can have different dimensions, but the number of columms of the first matrix is equal to the number of rows of the second matrix.
the product is a matrix with the same number of rows as the first matrix and the same number of columms as the second matrix.
the multiplication of matrices is not commutative, this means, in general is $A \cdot B \neq B \cdot A$
the multiplication of matrices is associative, this means $(A \cdot B)\cdot C = A\cdot(B\cdot C)$

[edit] Special matrices

There are some matrices that are special.

[edit] Square matrix

A square matrix has the same number of rows as columns, so m=n.

An example of a square matrix is

$\begin{bmatrix} 5 & -2 & 4 \\ 0 & 9 & 1 \\ -7 & 6 & 8 \\ \end{bmatrix}$

This matrix has 3 rows and 3 columns: m=n=3.

[edit] Identity matrix

Every dimension set of a matrix has a special counterpart called an "identity matrix". The identity matrix has nothing but zeroes except on the main diagonal, where there are all ones. For example:

$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$

is an identity matrix. There is exactly one identity matrix for each dimension.

[edit] Inverse matrix

An inverse matrix is a matrix that, when multiplied by another matrix, equals the identity matrix. For example:

$\begin{bmatrix} 7 & 8 \\ 6 & 7 \\ \end{bmatrix} \cdot \begin{bmatrix} 7 & -8 \\ -6 & 7 \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}$

$\begin{bmatrix} 7 & -8 \\ -6 & 7 \\ \end{bmatrix}$ is the inverse of $\begin{bmatrix} 7 & 8 \\ 6 & 7 \\ \end{bmatrix}$ .