The column rank of a matrix is the dimension of the linear space spanned by its columns.
The row rank of a matrix is the dimension of the space spanned by its rows.
Since we can prove that the row rank and the column rank are always equal, we simply speak of the rank of a matrix.
Let us start with a definition.
Definition
Let
be a
matrix. The column rank of
is![[eq1]](/images/rank-of-a-matrix__4.png){kind=icon}
where
denotes the
-th
column of
,
denotes the linear span, and
denotes the
dimension.
Remember that the dimension of a linear space is the number of elements of one of its bases, that is, the number of linearly independent vectors that generate the space. So, the column rank of a matrix is the number of linearly independent vectors that generate the same space generated by the columns of the matrix.
Example
Consider the matrix
and
the linear space
spanned by its two
columns![[eq3]](/images/rank-of-a-matrix__12.png){kind=icon}
that
is, the space of all vectors that can be written as
linear combinations of
and
.
Any vector
can be written
aswhere
and
are two scalars. Note that the two columns
and
are linearly dependent
becauseTherefore,
any vector
can be written as a multiple of
:
As
a consequence,
is
a basis for
.
It has
element. Therefore, the dimension of
and the column rank of
are equal to
.
The definition of row rank is analogous to that of column rank.
Definition
Let
be a
matrix. The row rank of
is![[eq8]](/images/rank-of-a-matrix__34.png){kind=icon}
where
denotes the
-th
row of
,
denotes the linear span, and
denotes the dimension.
In other words, the row rank of a matrix is the dimension of the linear space generated by its rows.
An important result is that the column rank of a matrix is always equal to its row rank.
Proposition
Let
be a
matrix.
Then,
Let
Then,
there exists a basis
of
column vectors that spans the same space spanned by the columns of
.
Denote by
the
matrix obtained from the vectors of the basis:
Each
column of
can be expressed as a linear combination of
.
The coefficients of the linear combinations can be collected into a
matrix
such
thatas
we have shown in the lecture on
Matrix
multiplication and linear combinations. In the same lecture, we have also
shown that the rows of the product
are linear combinations of the rows of
,
with coefficients taken from
.
So the span of the rows of
is no larger than the span of the rows of
(because linear combinations of the rows of
can be written as linear combinations of the rows of
). There
are
rows in
.
If they are linearly independent, then their span has dimension
.
Otherwise, it has dimension less than
.
As a consequence, the row rank of
is less than or equal to
(its column rank). In a completely analogous manner, we prove that the column
rank is less than or equal to the row rank: let
Then,
there exists a basis
of
row vectors that spans the same space spanned by the rows of
.
Denote by
the
matrix obtained from the vectors of the basis:
Each
row of
can be expressed as a linear combination of
.
The coefficients of the linear combinations can be collected into a
matrix
such
thatThe
columns of the product
are linear combinations of the columns of
,
with coefficients taken from
.
So the span of the columns of
is no larger than the span of the columns of
(because linear combinations of the columns of
can be written as linear combinations of the columns of
). There
are
columns in
.
If they are linearly independent, then their span has dimension
.
Otherwise, it has dimension less than
.
As a consequence, the column rank of
is less than or equal to
(its row rank). Thus, we have proved
thatand![[eq21]](/images/rank-of-a-matrix__94.png){kind=icon}
Therefore,
![[eq22]](/images/rank-of-a-matrix__95.png){kind=icon}
Having proved that column and row rank coincide, we are now ready to provide the definition of rank.
Definition
Let
be a
matrix. The rank of
,
denoted by
,
is defined as
![[eq24]](/images/rank-of-a-matrix__100.png){kind=icon}
In other words, the rank of a matrix is the dimension of the linear span of its columns, which coincides with the dimension of the linear span of its rows.
The following proposition holds.
Proposition
Let
be a
matrix.
Then
We have two possible cases. In the first
case,that
is, the number
of rows is less than or equal to the number
of columns. The columns are vectors having
entries. So, the dimension of the space spanned by the columns is less than or
equal to
.
In other
words,but
so
that![[eq29]](/images/rank-of-a-matrix__111.png){kind=icon}
In the second
case,that
is, the number
of columns is less than or equal to the number
of rows. The rows are vectors having
entries. So, the dimension of the space spanned by the rows is less than or
equal to
.
In other
words,but
so
that![[eq33]](/images/rank-of-a-matrix__119.png){kind=icon}
Thus,
in both
cases,
Given the previous results, we can now give a definition of full-rank matrix.
Definition
Let
be a
matrix. Then
is said to be full-rank if and only
if
Clearly, if
is a square matrix, that is, if
,
then it is full-rank if and only
ifIn
other words, if
is square and full-rank, then its columns (rows) span the space of all
-dimensional
vectors: any
-dimensional
vector can be written as a linear combination of the columns (rows) of
.
Below you can find some exercises with explained solutions.
Let
be a
matrix. What is the maximum rank that it can have?
The maximum rank of
is
Define
What is the rank of
?
The matrix
has two
columns:![[eq39]](/images/rank-of-a-matrix__139.png){kind=icon}
The
two columns are linearly independent because neither of them can be written as
a scalar multiple of the other. As a matter of fact, they are not
multiples. This can be clearly seen from the third entry of
which is
:
there is no coefficient that can be multiplied by
to obtain
,
the third entry of
.
Therefore, the span of the columns of
has dimension
,
that is, the column rank of
is equal to
,
and
Please cite as:
Taboga, Marco (2021). "Rank of a matrix", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/rank-of-a-matrix.
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