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.
![$\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}\dot{=}a_{11} a_{22} - a_{21} a_{12}$](http://latex.artofproblemsolving.com/b/2/2/b222819415371df638b5204d107f82ca17aa6496.png)
![$\begin{vmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn}\end{vmatrix}\dot{=}\sum_{k=1}^n (-1)^{k+1} a_{1k} |A'_{1k}|$](http://latex.artofproblemsolving.com/c/b/c/cbc205d27b592353c4fed8c0340301805f8adf31.png)
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
.