or
= | x_{1 } | |||
x_{2 } | ||||
: | ||||
x_{n } | ||||
These two are special cases of a matrix
A = | a_{11 } | a_{12 } | ··· | a_{1n } | . | ||
a_{21 } | a_{22 } | ··· | a_{2n } | ||||
: | : | : | |||||
a_{m1 } | a_{m2 } | ··· | a_{mn } | ||||
The numbers a_{ij } are called the entries or the elements of the matrix A.
Note that in the double-subscript notation for the elements, the subscript i denotes the
row while j denotes the
column in which the element a_{ij } stands.
The dimension or the size of the above matrix A is m x n.
One often writes A_{m} _{x n} or
If m = n then A is a square matrix.
We also say that a square matrix A is
1) an upper triangular matrix, if a_{ij } = 0 when i > j.
x | x | x | x | x | ||
0 | x | x | x | x | ||
0 | 0 | x | x | x | ||
0 | 0 | 0 | x | x | ||
0 | 0 | 0 | 0 | x | ||
2) a lower triangular matrix, if a_{ij } = 0 when i < j.
x | 0 | 0 | 0 | 0 | ||
x | x | 0 | 0 | 0 | ||
x | x | x | 0 | 0 | ||
x | x | x | x | 0 | ||
x | x | x | x | x | ||
3) a diagonal matrix, if a_{ij } = 0 when i j.
x | 0 | 0 | 0 | 0 | ||
0 | x | 0 | 0 | 0 | ||
0 | 0 | x | 0 | 0 | ||
0 | 0 | 0 | x | 0 | ||
0 | 0 | 0 | 0 | x | ||
Only the elements on the main diagonal can be 0.
4) A tridiagonal matrix, if a_{ij } = 0 when | i - j | > 1.
x | x | 0 | 0 | 0 | ||
x | x | x | 0 | 0 | ||
0 | x | x | x | 0 | ||
0 | 0 | x | x | x | ||
0 | 0 | 0 | x | x | ||
5) An upper Hessenberg matrix, if a_{ij } = 0 when i > j + 1.
A lower Hessenberg matrix, if a_{ij } = 0 when j > i + 1.
6) A Toeplitz matrix if a_{ij } = a( i - j ), for example
a(0) | a(-1) | a(-2) | a(-3) | ||
a(1) | a(0) | a(-1) | a(-2) | ||
a(2) | a(1) | a(0) | a(-1) | ||
a(3) | a(2) | a(1) | a(0) | ||
A diagonal matrix is often written as
_{1 } | 0 | 0 | 0 | 0 | = diag (_{1 }, _{2 }, _{3 }, _{4 }, _{5 }). | ||
0 | _{2 } | 0 | 0 | 0 | |||
0 | 0 | _{3 } | 0 | 0 | |||
0 | 0 | 0 | _{4 } | 0 | |||
0 | 0 | 0 | 0 | _{5 } | |||
The identity matrix of dimension n, denoted by I_{n } (or I, if the dimension is clear from the context), is
I_{n } = diag (1, 1, ..., 1) = | 1 | 0 | 0 | 0 | ··· | 0 | ||
0 | 1 | 0 | 0 | ··· | 0 | |||
0 | 0 | 1 | 0 | ··· | 0 | |||
: | ||||||||
0 | 0 | ··· | 0 | 1 | ||||
I = (_{ij }) , | _{ij } = | 1, | kun | i = j | |
0, | kun | i j | |||
A zero matrix O (or O_{m}_{ x n }) is a matrix whose elements are all zeros.