Eigenvalue decomposition plays a very vital role in easing the complexity of the matrix in the Linear Algebra field. The square matrix is broken down into simple components, i.e., eigenvalues and their eigenvectors. In different machine learning and deep learning models, we need to deal with the square matrix. This matrix needs to be sorted, manipulated, and analyzed perfectly. So, eigenvalue decomposition helps to solve this problem. In this article, we will see the details of the eigenvalues decomposition and its implementation in Python.
The eigenvalue decomposition is a very important concept in linear algebra to perform various tasks. The different operations involve the diagonal conversion of a square matrix. The diagonal matrix is very easy to operate because all the elements are zero except the diagonal part. This method automatically minimizes the complexity of the matrix problem in Python. This eigenvalue decomposition is also used to find out the PCA, i.e., Principal Component Analysis. The Principal Component Analysis is used in the dimensionality reduction technique, which is widely used in machine learning algorithms.
What is the Eigenvalue and Eigenvector in Python?
The eigenvalue is a scalar value that is used to multiply with the eigenvector, which helps to find out the diagonal values of the matrix. Eigenvalue is the scalar term that represents the transformation of the matrix. The eigenvector is a non-zero vector. Both eigenvalues and eigenvectors are part of a technique called eigenvalue decomposition. The square matrix can be represented in the form of an eigenvalue and an eigenvector. Let’s see the mathematical representation of the eigenvalue decomposition.
The formula for decomposing eigenvalues can be shuffled around sporadically to still convey the same information, but make it more distinguishable. In diverse machine learning and deep learning models, we can use a distinct formula containing a square matrix A, an eigenvalue represented by λ, and a non-zero eigenvector, v.
Significance of Diagonalization in Linear Algebra
Diagonalization is a very interesting technique in the linear algebra domain. The eigenvalue decomposition and diagonalization are both related to the eigenvalues and eigenvectors, but there is a clear difference between the two terms. For example, diagonalization helps to find the diagonal matrix from the square matrix, and eigenvalue decomposition helps to find the three different matrices. There are different applications of diagonalization in Python. First, try to understand the mathematical formula behind the diagonalization. In linear algebra, the diagonalization process is used in various models. The diagonalization helps to decompose complex problems into simple steps.
The diagonalization is represented as a product of the invertible matrix P, A, and P. P is the matrix formed by the eigenvectors of matrix A. There are several benefits to using the concept of diagonalization in linear algebra and different domains to solve problems.
The diagonalization of the matrix helps to decrease the overall complexity of calculation in the model. The matrix multiplication process becomes much easier and involves a diagonal matrix. In machine learning and data analysis, domain diagonalization plays a vital role in finding out the eigenvalues and eigenvectors that are used in a Principal Component Analysis (PCA). This PCA technique is used for dimensionality reduction in different models. Diagonalization can be used to transform a complex model into a simpler one. The problems of network theory can be solved using the process of diagonalization.
Steps Involved in Eigenvalue Decomposition
There are a few steps involved in the process of eigenvalue decomposition. For eigenvalue decomposition, we need to start with the square matrix, assuming A. The next step involves the calculation of the eigenvalue and eigenvector. After calculating the eigenvalue and eigenvectors, check the diagonalizability of the matrix. Then, the inverse of the matrix is calculated. In this way, we can calculate the eigenvalue decomposition. Let’s try to implement this using Python code.
Eigenvalue Decomposition in Python
The implementation of eigenvalue decomposition can be facilitated in a variety of models through the two aforementioned Python libraries. Specifically, the libraries in question are scipy and numpy. While scipy employs the eig() function, numpy utilizes the numpy. linalg.eig() function. To begin deploying these libraries for implementation purposes, first ensure that they are both properly installed.
Eigenvalue Decomposition Using Numpy
In Python, create eigenvalue decomposition with the assistance of the numpy library through its import and installation.
pip install numpy
After installation of the Numpy library, let’s try to implement further code!
import numpy as np
A = np.array([[3, 1], [1, 3]])
eigenvalues, eigenvectors = np.linalg.eig(A)
V = eigenvectors
Λ = np.diag(eigenvalues)
V_inv = np.linalg.inv(V)
A_decomposed = np.dot(np.dot(V, Λ), V_inv)
print("Original Matrix A:\n", A)
print("Eigenvalue Decomposition of A:\n", A_decomposed)
A numpy library import is present within the code. An A square matrix is defined, and the calculations of eigenvectors and eigenvalues of said matrix stem from the np. linalg.eig() function from the numpy library. As elaborated on in previous sections, eigenvector inversion is computed and utilized in the overall formula for eigenvalue decomposition.
In the end, we will print the result with the original matrix and the new matrix. Let’s see the output.
The original matrix has the same eigenvalues and diagonal elements; therefore, the output is the same matrix. In this way, we can calculate the eigenvalues, eigenvectors, and eigenvalue decomposition for every square matrix that is diagonalizable.
Eigenvalue Decomposition Using Scipy. linalg
In this implementation, let’s try to install the scipy. linalg. This library is specifically built for calculations related to linear algebra. This library is used under the numpy.linalg library. This library provides different functions for problems related to linear algebra, like eigenvalue decomposition, linear algebra equations, matrix factorization, etc.
pip install scipy
Let’s see the full code for implementation.
import numpy as np
from scipy.linalg import eig
A = np.array([[3, 1], [1, 3]])
eigenvalues, eigenvectors = eig(A)
V = eigenvectors
Λ = np.diag(eigenvalues)
V_inv = np.linalg.inv(V)
A_decomposed = np.dot(np.dot(V, Λ), V_inv)
print("Original Matrix A:\n", A)
print("Eigenvalue Decomposition of A:\n", A_decomposed)
In this code, the same process is used as a numpy library. First, we installed and imported the Scipy library. Here, we are using the eig() function from the scipy library, which is considered an alternative to the np. linalg.eig() function of numpy. Let’s see the output of this code.
What is the Use of Eigenvalue Decomposition?
There is a wide range of applications for eigenvalue decomposition due to its ability to convert complex mathematical operations into simple ones. The Principal Component Analysis (PCA) uses eigenvalue decomposition for dimensionality reduction techniques in the Python language. Many machine learning and deep learning models require this PCA technique to reduce the dimensions of the datasets that involve images and signals. Another application of this technique is image compression.
Image compression is mainly used in machine learning and deep learning. There is a technique called Singular Value Decomposition (SVD) that is used for image compression in Python and implements eigenvalue decomposition. The data analysis field also uses this technique to analyze large amounts of data, like seismic and financial data. In this case, the data needs to be optimized or image compression techniques need to be implemented on the whole dataset. In this case, we use this technique to implement the model accurately.
Limitations of Eigenvalue Decomposition
In many machine learning and deep learning algorithms, we require eigenvalue decomposition techniques for various applications. Sometimes, we face some limitations while implementing this technique. There are some basic requirements for this technique, like the fact that the formula is only applicable to the square matrix. If the matrix is non-square, then this technique cannot be implemented. One more basic requirement of this formula is a diagonalizable matrix. The non-diagonalizable matrix cannot be decomposed into eigenvalues and eigenvectors. Therefore, the diagonalizable matrix is very important.
This technique is not feasible for large matrices. For large matrices, calculations are very complex and time-consuming. The memory required to store these eigenvalues and eigenvectors is also high. This may lead to many errors. Therefore, this technique is best for small-dimension matrices.
Summary
In this article, the eigenvalue decomposition technique is explained in detail. This technique is useful in many machine learning and deep learning models. The square matrix can be converted into eigenvalues and eigenvectors. The single matrix can be represented in the form of three different matrices. The mathematical formula and theoretical explanation of the general formula are also explained in detail.
There is one more technique that is part of eigenvalue decomposition and is also known as the process of diagonalization. Diagonalization is the conversion of the square matrix into a diagonal matrix. The difference between these two techniques is explained in detail with the help of mathematical formulas and an explanation.
In the next section, an example based on the implementation of eigenvalue decomposition using two different libraries is given. The first example uses the numpy library, and the second example is implemented using the scipy library in Python. The detailed implementation of both examples is given; therefore, we can understand the concept of eigenvalue decomposition clearly.
In the end, the real-world application and limitations of this technique are also explained in detail. There is a wide range of applications, and some fundamental limitations need to be considered while implementing the technique in different models. I hope you will enjoy this article!
Also, check this article on Hot to Compute the eigenvalues of a general matrix?
References
Read the official documentation of the numpy and scipy libraries.