# Matrix

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 .

## Contents

## Determinant

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.

## Transposes

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 .

## Matrix Product

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 .