Rank of a matrix

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.

Column rank

Definition Let be a matrix. The column rank of iswhere 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 columnsthat 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 .

Row rank

The definition of row rank is analogous to that of column rank.

Definition Let be a matrix. The row rank of iswhere 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.

Column rank equals row rank

An important result is that the column rank of a matrix is always equal to its row rank.

Proposition Let be a matrix. Then,

Proof

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 thatandTherefore,

Definition of rank

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

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.

Maximum rank

The following proposition holds.

Proposition Let be a matrix. Then

Proof

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 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 thatThus, in both cases,

Full-rank

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 .

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let be a matrix. What is the maximum rank that it can have?

Solution

The maximum rank of is

Exercise 2

Define What is the rank of ?

Solution

The matrix has two columns: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