# MatrixRank

MatrixRank[m]

gives the rank of the matrix m.

# Details and Options • MatrixRank works on both numerical and symbolic matrices.
• The rank of a matrix is the number of linearly independent rows or columns.
• MatrixRank[m,Modulus->n] finds the rank for integer matrices modulo n.
• MatrixRank[m,ZeroTest->test] evaluates test[m[[i,j]]] to determine whether matrix elements are zero. The default setting is .
• MatrixRank[m,Tolerance->t] gives the minimum rank with each element in a numerical matrix assumed to be correct only to within tolerance t.
• MatrixRank works with sparse arrays.

# Examples

open allclose all

## Basic Examples(1)

Find the number of linearly independent rows:

## Scope(5)

MatrixRank can handle rectangular matrices:

MatrixRank assumes all symbols to be independent:

MatrixRank for a machinenumber matrix:

Find the matrix rank for a complexvalued matrix:

Find the rank of a sparse matrix:

## Options(2)

### Modulus(1)

The rank of a matrix depends on the modulus used:

With ordinary arithmetic, m has the full rank of 3:

With arithmetic modulo 5, the rank is only 2:

### Tolerance(1)

The setting of Tolerance can affect the estimated rank for numerical ill-conditioned matrices:

In exact arithmetic, m has full rank:

With machine arithmetic, the default is to consider elements that are too small as zero:

With zero tolerance, even small terms may be taken into account:

With a tolerance greater than the pivot in the middle row, the last two rows are considered zero:

## Applications(2)

Most but not all random 10×10 01 matrices have full rank:

Estimate the average rank of a random 10×10 01 matrix:

Find the ranks of coprimality arrays:

## Properties & Relations(4)

MatrixRank[m] is equal to :

MatrixRank[m] plus the dimension of the null space is equal to the number of columns of m:

The column and row rank of a matrix are equal:

The outer product of vectors has matrix rank 1:

## Possible Issues(1)

MatrixRank may depend on the precision of the given matrix:

Use exact arithmetic to compute the matrix rank exactly:

Use machine arithmetic. Machine numbers cannot distinguish between and :

Use 24digit-precision arithmetic:

Introduced in 2003
(5.0)
|
Updated in 2007
(6.0)