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. Matrices are also widely used in computer science.

Contents

[change] 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 aij.

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

[change] 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.

[change] Operations

[change] 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.

[change] Multiplication of two matrices

The multiplication of two matrices is a bit more complicated:

\begin{bmatrix} a1 & a2 \\ a3 & a4 \\ \end{bmatrix} \cdot \begin{bmatrix} b1 & b2 \\ b3 & b4 \\ \end{bmatrix} = \begin{bmatrix} (a1\cdot b1 + a2 \cdot b3) & (a1 \cdot b2 + a2 \cdot b4) \\ (a3\cdot b1 + a4 \cdot b3) & (a3 \cdot b2 + a4 \cdot b4) \\ \end{bmatrix}
  • two matrices can have different dimensions, but the number of columns 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 columns 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)

[change] Special matrices

There are some matrices that are special.

[change] 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.

[change] 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.

[change] 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}.

[change] One column matrix

A matrix, that has many rows, but only one column, is called a column vector.

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