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Linear Algebra for Data Science Using Python

Inverse of a Rectangular Matrix

Explore the inverse of rectangular matrices and the idea of left and right inverses.

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We’ve already discussed the inverse of a square matrix in detail and provided multiple examples. In this lesson, we’ll extend the discussion to rectangular matrices, which are matrices that have a different number of columns and rows.

Rank and invertibility

The rank of a matrix generally answers the question of invertibility. An m×nm \times n matrix AA is invertible if it has either a full column rank or a full row rank.

r(A)=min(m,n)r(A) = min(m,n)

Also, for any matrix, AA,

r(A)=r(ATA)=r(AAT)r(A)=r(A^TA)=r(AA^T)

Python 3.8
import numpy as np
from numpy.linalg import matrix_rank as r
order = np.random.randint(low=2, high=10, size=2)
m, n = order[0], order[1]
A = np.random.rand(m, n)
print(f'order(A)={A.shape}\nr(A)={r(A)}\nr(ATA)={r(A.T.dot(A))}\nr(AAT)={r(A.dot(A.T))}')

For square matrices, the number of rows, mm, is equal to the number of columns, nn. This allows square matrices to have a two-sided inversetwosided_Inverse. For rectangular matrices, the condition doesn’t hold. They either have a left inverse or a right inverse.

Left inverse

If a matrix, AA, has full column rank, r(A)=nr(A)=n, then it would have a left inverse, LL, such that:

A matrix with a full column rank could have multiple left inverses.

One possibility is:

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