matrix rank

Topics

• linear algebra
• matrices

The rank of a matrix is an estimate of the number of linearly independent rows or columns in the matrix. It is denoted as $\text{rank}(A)$. The rank provides an intuition about the dimensionality of the vector space spanned by the vectors within the matrix:

• a rank of 1 suggests the vectors span a line
• a rank of 2 suggests they span a plane, etc

The rank is typically estimated numerically, using the Singular-Value Decomposition (SVD).

The rank indicates the number of linearly independent directions within the matrix, not necessarily the number of dimensions of the matrix itself

from numpy.linalg import matrix_rank
 
A0 = np.zeros((3,3))
r0 = matrix_rank(A0)  # 0
 
A1 = np.array([[1,2,3],
               [1,2,3],
               [1,2,3]])
r1 = matrix_rank(A1)  # 1
 
A2 = np.eye(3)
r2 = matrix_rank(A2)  # 3

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