rank

Matrix Rank

Usages

rank(A)

• It returns the rank of a numeric (non-character) matrix A not containing Inf nor NaN.
• More specifically, it returns the number of singular values of A larger than the default tolerance:

max(size(A)) * eps(norm(A, 2))

• It returns 0 if A is empty.

rank(A, tol)

• A is the same as specified above for rank(A).
• tol is an array containing tolerances, which must be real non-negative numbers.
• The output array has the same size as tol.
• Each element in the output array is the number of singular values of A larger than the corresponding tolerance in tol.
• If a tolerance in tol is inf or nan, the corresponding number in the output array is zero.
• If A is empty, it returns an array of all zeros, regardless of the tolerances.

Examples

• A below has rank 2.

Input

A = [1 0 0 0;0 1 0 0;0 1 0 0];
rank(A)

Output

ans =
 2.000

• Using the default tolerance tol = max(size(A))*eps(norm(A, 2)), rank(A) and rank(A,tol) give the same results.

Input

A = [1 2 3 4;1 2 3 4;0 1 0 0]
r1 = rank(A)
tol = max(size(A))*eps(norm(A, 2))
r2 = rank(A,tol)

Output

A =
 1.000   2.000   3.000   4.000
 1.000   2.000   3.000   4.000
 0.000   1.000   0.000   0.000

r1 =
 2.000

tol = 1e-15 ×
 3.553

r2 =
 2.000