Matrices are denoted by upper case letters such as A, B, C, ...

A matrix with m rows and n columns is called an m x n matrix (pronounce m-by-n matrix).

Examples

The element on the i-th row and j-th column of a matrix A is denoted by ai,j. The subscripts indicate the row first and the column second.

A matrix in the general form is written as follows:

A short form of this equality is

.

The size of the matrix is given by the number of rows and the number of columns.

If both matrices, A and B, are m x n matrices, then they have the same size.

A matrix with one row is called a row matrix.

For instance, the following one-by-three column matrix

is a row matrix.

A matrix with one column is called a column matrix:

A matrix is called a square matrix if the number of its rows equals the number of the columns.

A square matrix of n-th order is a n x n matrix.

In a square matrix the elements ai,i, with i = 1, 2, 3, ..., are called diagonal matrix elements.

A matrix is called a diagonal matrix if all its off-diagonal elements are equal to zero, but at least one of the diagonal elements is nonzero:

Examples

1. The diagonal of the following square matrix of the third order

consists of numbers 4, 5, and 9, which are diagonal elements.

2. The matrix

is a diagonal matrix of the third order.

An identity matrix is a diagonal matrix whose diagonal elements are equal to unity.

Usually, identity matrices are denoted by the symbol I. The matrix elements of an identity matrix I are denoted by the symbol which is known as the Kronecker delta symbol.

Thus,

,

where

The delta symbol cancels symmation over one of the symbols in such expressions as

Each of the above sum contains only one nonzero item, e.g.,

.

All the rest entries are equal to zero, since for any .

For instance, if j = 3 then , while , and so

.

Examples

The diagonal subdivides a square matrix into two blocks, one above the diagonal and the other one below it. The matrix has a triangular form, if one of the blocks consists of zeros:

all ai,j=0 for i>j or for i<j

Examples

Given a matrix , the transpose of A is the matrix obtained from A by interchanging its rows and columns. This means that the rows of a matrix A are the columns of the matrix ; and vise versa:

.

Example

If then .

A square matrix is called symmetric if

.

Example

The following matrix

is a symmetric matrix, since .

A square matrix is called skew-symmetric if

A matrix is called a zero-matrix (or 0-matrix) if all its elements are equal to zero, that is,

for each pair of indexes {i, j}.

In a short form 0-matrix is written as 0.