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Python for Scientists and Engineers

Matrix Operations

Explore key matrix operations used in solving systems of linear equations including calculating the inverse, determinant, trace, transpose, and Euclidean norm of matrices using Python. Understand how these operations assist in efficiently obtaining solutions to linear problems with practical implementation examples.

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Inverse

Suppose we have to solve the equation AX=bAX=b for bb.

If vector bb changes, one has to call np.linalg.solve() over and over. Instead, if A−1A^{-1} were calculated once, matrix multiplication with bb could be done over and over, more efficiently, to obtain different solutions.

The inverse of a matrix may be computed with the inv function of the linalg package.

Let’s use the inverse of the matrix to obtain the solution to the problem in the first lesson.

AX=bAX=b

A−1AX=A−1bA^{-1}AX=A^{-1}b

X=A−1bX=A^{-1}b

If the inverse of matrix A is called Ainv, the solution for the equation may be obtained through matrix multiplication of Ainv with the right-hand side. Let’s look at an implementation of this below.

Python 3.5
import numpy as np
A = np.array([[2, 3, 1], [3, 4, 2], [1, 1, -1]])
b = np.array([17, 25, 6])
Ainv = np.linalg.inv(A)
x =  np.dot(Ainv, b)
print(x)

Determinant

det() from the linalg submodule is used to compute the determinant of the input matrix.

Python 3.5
import numpy as np
A = np.array([[2, 3, 1], [3, 4, 2], [1, 1, -1]])
print(np.linalg.det(A))

Trace

trace() from the numpy module is used to compute the trace of the input matrix.

Trace is the sum of the diagonal values of a matrix.

Python 3.5
import numpy as np
A = np.array([[2, 3, 1], [3, 4, 2], [1, 1, -1]])
print(np.trace(A))

Transpose

transpose() from the numpy module returns the transpose of the input array.

The transpose of a matrix is a new matrix whose rows are the columns of the original.

Python 3.5
import numpy as np
A = np.array([[2, 3, 5], [4, 98, 2], [11, 12, -13]])
print(A)
print("After Transpose")
print(np.transpose(A))

Euclidean norm

The Euclidean norm of an m×nm \times n matrix is given by the following equation:

∥A∥=∑i=1m∑j=1n∣aij∣2\parallel A \parallel =\sqrt{\sum_{i=1}^m\sum_{j=1}^n\mid a_{ij}\mid^2}

The norm() method from the linalg submodule is used to calculate the Euclidean norm of a matrix.

Python 3.5
import numpy as np
v = np.array([0, 3, 4])
M = np.array([[2, 3, 5], [4, 98, 2], [11, 12, -13]])
print("Norm of v is", np.linalg.norm(v))
print("Norm of M is", np.linalg.norm(M))

In the next lesson, we will discuss sparse matrices.