Orthogonal matrix is an important matrix in linear algebra, it is also widely used in machine learning. In this tutorial, we will dicuss what it is and how to create a random orthogonal matrix with pyhton.
What is Orthogonal Matrix?
If a matrix A is an orthogonal matrix, it shoud be n*n.
The feature of an orthogonal matrix A.
The euclidean length of A.
How to create a random orthogonal matrix?
The simplest orthogonal matrix is one-hot encoding, such as:
[[1, 0, 0] [0, 1, 0] [0, 0, 1]]
However, the value in it is not random. How to create a random orthogonal matrix?
Here is an simple example, we will use python scipy to implement it.
from scipy.stats import ortho_group # Requires version 0.18 of scipy import numpy as np m = ortho_group.rvs(dim=5) print(m)
Here we will create a 5 * 5 random orthogonal matrix, it is:
[[-0.04861857 -0.44507735 -0.38079495 0.31292116 -0.74606833] [-0.20933804 0.4058631 0.35547015 -0.52018465 -0.62809365] [ 0.53353666 0.63968878 -0.53749448 0.05881791 -0.11737561] [ 0.45728819 0.08815114 0.66040851 0.55928113 -0.18488401] [ 0.67826246 -0.46926426 0.05997047 -0.56145645 -0.03035287]]
We check it is an orthogonal matrix or not.
l1 = np.matmul(m, m.T) print(l1) l2 = np.matmul(m.T, m) print(l2)
The result is:
[[ 1.00000000e+00 7.84994566e-17 1.65829696e-16 -1.31158853e-16 -9.57636165e-18] [ 7.84994566e-17 1.00000000e+00 -1.98313914e-16 1.25646971e-16 5.00488907e-17] [ 1.65829696e-16 -1.98313914e-16 1.00000000e+00 -9.72148193e-17 -2.25065344e-17] [-1.31158853e-16 1.25646971e-16 -9.72148193e-17 1.00000000e+00 9.59854042e-17] [-9.57636165e-18 5.00488907e-17 -2.25065344e-17 9.59854042e-17 1.00000000e+00]] [[ 1.00000000e+00 8.75665129e-17 -1.39245100e-16 1.47515708e-16 6.06719417e-17] [ 8.75665129e-17 1.00000000e+00 -4.07319982e-17 1.76818401e-17 -5.21988496e-17] [-1.39245100e-16 -4.07319982e-17 1.00000000e+00 -1.89462221e-16 2.42141102e-17] [ 1.47515708e-16 1.76818401e-17 -1.89462221e-16 1.00000000e+00 1.42728087e-16] [ 6.06719417e-17 -5.21988496e-17 2.42141102e-17 1.42728087e-16 1.00000000e+00]]
From the result, we will find the matrix m is a random orthogonal matrix.