MatrixRank

MatrixRank can handle rectangular matrices:

MatrixRank assumes all symbols to be independent:

MatrixRank for a machine-number matrix:

Find the matrix rank for a complex-valued matrix:

Find the rank of a sparse matrix:

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:

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:

Most but not all random 10×10 0-1 matrices have full rank:

Estimate the average rank of a random 10×10 0-1 matrix:

Find the ranks of coprimality arrays:

MatrixRank[m] is equal to Length[SingularValueList[m]]:

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 have matrix rank 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 24-digit precision arithmetic: