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Linear Algebra for Data Science Using Python

Elementary Matrices

Explore how elementary matrices correspond to fundamental row operations such as swaps, scaling, and sums. Understand how to construct these matrices from identity matrices, use them to transform and decompose matrices, and combine them into elimination matrices for efficient processing. Learn to compute inverses of these matrices to reverse operations, gaining core skills for matrix manipulations in data science contexts.

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An elementary matrix is a square matrix that represents an elementary row operation. These matrices help us automate matrix transformations and decompositions.

Creation of an elementary matrix

To create an elementary matrix, En×nE_{n \times n}, we start with the identity matrix, In×nI_{n\times n}, and apply the desired elementary row operation to it. The resultant matrix is the elementary matrix corresponding to the given operation.

Row swap

An elementary matrix for row swap is an identity matrix with the two rows interchanged. These matrices are also called permutation matrices, represented by PijP_{ij}, where ii and jj represent the two rows to be swapped. For example, the elementary matrix to swap the second and third rows is as follows:

P23=[100001010]P_{23}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0 \end{bmatrix} ...

Python 3.8
import numpy as np
I = np.eye(3, 3, dtype=np.int32)
print(f"""Identity matrix:\n {I}""")
I[[1, 2], :] = I[[2, 1], :]
E = I
print(f"""Elementary matrix:\n {E}""")

Row scaling

An elementary matrix to scale (up or down) the ...