A matrix over a field is a function from to , where and are the sets and . A matrix is usually represented as a rectangular array of scalars from the field, such that each column belongs to the vector space , where is the number of rows. If a matrix has rows and columns, its order is said to be , and it is written as .
The element in the row and column of is written as . It is more often written as , in which case can be written as .
If is a matrix over with , a Determinant assigns to a member of and is denoted by or
It is defined recursively.
where is the matrix with the row and column removed.
Let be . Then is said to be the transpose of , written as or simply . If A is over the complex field, replacing each element of by its complex conjugate gives us the conjugate transpose of . In other words,
is said to be symmetric if and only if . is said to be hermitian if and only if . is said to be skew symmetric if and only if . is said to be skew hermitian if and only if .
Let be a matrix of order and a matrix of order . Then the product exists if and only if and in that case we define the product as the matrix of order for which for all and such that and .
Vector spaces associated with a matrix
As already stated before, the columns of form a subset of . The subspace of generated by these columns is said to be the column space of , written as . Similarly, the transposes of the rows form a subset of the vector space . The subspace of generated by these is known as the row space of , written as .
implies such that
Similarly, implies such that
The set forms a subspace of , known as the null space of .
Rank and nullity
The dimension of is known as the column rank of . The dimension of is known as the row rank of . These two ranks are found to be equal, and the common value is known as the rank of .
The dimension of is known as the nullity of A.
If is a square matrix of order , then .