Eigenvalues and Eigenvectors | Control Systems - Electrical Engineering (EE) PDF Download

Introduction

Eigenvectors and eigenvalues are core concepts in linear algebra for square matrices and linear transformations. An eigenvector is a non-zero vector that, under a matrix transformation, only scales by a factor, remaining parallel to its original direction. This scaling factor is the eigenvalue, satisfying $Av=\lambda vA\; v\; =\; \backslash lambda\; v$, where $AA$ is the matrix, $vv$ is the eigenvector, and $\lambda \backslash lambda$ is the eigenvalue.

Eigenvectors maintain their direction during transformation, while eigenvalues measure the scaling. They are vital in fields like engineering for stability analysis, quantum mechanics for energy states, and machine learning for dimensionality reduction (e.g., in Principal Component Analysis). These concepts are essential for solving problems in mathematics, physics, and computational disciplines.

Eigenvalues

Eigenvalues are the scalar values associated with the eigenvectors in linear transformation. The word ‘Eigen’ is of German Origin which means ‘characteristic’.

Hence, these characteristic values indicate the factor by which eigenvectors are stretched in their direction. It doesn’t involve the change in the direction of the vector except when the eigenvalue is negative. When the eigenvalue is negative the direction is just reversed.

The equation for eigenvalue is given by

Av = λv

Where,

A is the matrix,

v is associated eigenvector, and

λ is scalar eigenvalue.

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Eigenvectors

Eigenvectors of a square matrix are non-zero vectors that, when multiplied by the matrix, result in a scaled version of the same vector. For a square matrix $AA$A, I define an eigenvector $vv$v as one that satisfies the condition $Av=\lambda vAv\; =\; \backslash lambda\; v$Av=λv, where $\lambda \backslash lambda$λ is a scalar. This scalar $\lambda \backslash lambda$λ is called the eigenvalue of the matrix. To find the eigenvectors, I first need to determine the eigenvalues of the square matrix.

For a square matrix $AA$A of order $n\times nn\; \backslash times\; n$n×n, I understand the eigenvector to be a column matrix of order $n\times 1n\; \backslash times\; 1$n×1. When I find the eigenvector using the equation $Av=\lambda vAv\; =\; \backslash lambda\; v$Av=λv, the vector $vv$v is called the right eigenvector of matrix $AA$A. It’s always multiplied on the right-hand side because matrix multiplication isn’t commutative. Generally, when I refer to an eigenvector, I mean the right eigenvector.

I can also find the left eigenvector of the square matrix $AA$A using the relation $vA=v\lambda vA\; =\; v\backslash lambda$vA=vλ. Here, $vv$v is the left eigenvector and is multiplied on the left-hand side. For a matrix $AA$A of order $n\times nn\; \backslash times\; n$n×n, this left eigenvector $vv$v is a row matrix of order $1\times n1\; \backslash times\; n$1×n.

The Eigenvector equation is the equation that is used to find the eigenvector of any square matrix. The eigenvector equation is,

Av = λv

Where,

• A is the given square matrix,
• v is the eigenvector of matrix A, and
• λ is any scaler multiple.

What are Eigenvalues and Eigenvectors?

For a square matrix $AA$A of order $n\times nn\; \backslash times\; n$n×n, I can find its eigenvectors using the method below:

I know that an eigenvector satisfies the equation $Av=\lambda vAv\; =\; \backslash lambda\; v$Av=λv. To solve this, I use the identity matrix $II$I of the same order as $AA$A, which is $n\times nn\; \backslash times\; n$n×n, and rewrite the equation as:

$(A-\lambda I)v=0(A\; -\; \backslash lambda\; I)v\; =\; 0$(A−λI)v=0

By solving this equation, I obtain various values of $\lambda \backslash lambda$λ, which I label as $\lambda 1,\lambda 2,\dots ,\lambda n\backslash lambda\_1,\; \backslash lambda\_2,\; \backslash ldots,\; \backslash lambda\_n$λ1,λ2,…,λn. These values are called the eigenvalues, and each eigenvalue corresponds to its own eigenvector.

After simplifying the equation, I find $vv$v, which is a column matrix of order $n\times 1n\; \backslash times\; 1$n×1. The eigenvector $vv$v is written as:

To find the eigenvector of a square matrix, follow these steps:

Step 1: I start by finding the eigenvalues of the matrix $AA$A using the equation $\det (A-\lambda I)=0\backslash det(A\; -\; \backslash lambda\; I)\; =\; 0$det(A−λI)=0, where $II$I is the identity matrix of the same order as $AA$A.

Step 2: The values I obtain from Step 1 are the eigenvalues, which I label as $\lambda 1,\lambda 2,\lambda 3,\dots \backslash lambda\_1,\; \backslash lambda\_2,\; \backslash lambda\_3,\; \backslash ldots$λ1,λ2,λ3,….

Step 3: Next, I find the eigenvector $XX$X corresponding to the eigenvalue $\lambda 1\backslash lambda\_1$λ1 by solving the equation $(A-\lambda 1I)X=0(A\; -\; \backslash lambda\_1\; I)\; X\; =\; 0$(A−λ1I)X=0.

Step 4: I repeat Step 3 for the remaining eigenvalues $\lambda 2,\lambda 3,\dots \backslash lambda\_2,\; \backslash lambda\_3,\; \backslash ldots$λ2,λ3,…, to find their corresponding eigenvectors.

Types of Eigenvector

Types of Eigenvectors

When we calculate the eigenvectors of a square matrix, we find that they are of two types:

Right Eigenvector

The right eigenvector is defined as the one multiplied by the square matrix from the right-hand side. It is calculated using the equation:

$AVR=\lambda VRA\; V\_R\; =\; \backslash lambda\; V\_R$

Here, $AA$ is the given square matrix of order $n\times nn\; \backslash times\; n$, $\lambda \backslash lambda$ is one of the eigenvalues, and $VRV\_R$ is the column vector matrix. The form of $VRV\_R$ is: 

Left Eigenvector

The left eigenvector, in contrast, is multiplied by the square matrix from the left-hand side. It is calculated using the equation:

VLA=VLλ

$\mathrm{<br>}$Where,

A is given square matrix of order n×n,

λ is one of the eigenvalues, and

VL is the row vector matrix.

$V\_L\; A\; =\; V\_L\; \backslash lambda$The value of V_L is,

V_L = [v₁, v₂, v₃,…, v_n]

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Applications of Eigenvalues in Engineering

Eigenvalues have several key applications across various fields of engineering, including:

Linear Algebra

• Diagonalization: Eigenvalues enable the diagonalization of matrices, which simplifies computations and enhances the efficiency of solving linear systems.
• Matrix Exponentiation: Eigenvalues are essential for computing the exponentiation of a matrix, facilitating complex matrix operations.

Quantum Mechanics

• Schrödinger Equation: Eigenvalues of the Hamiltonian operator correspond to the energy levels of quantum systems, revealing the possible states of the system.

Vibrations and Structural Analysis

• Mechanical Vibrations: Eigenvalues represent the natural frequencies of vibrational systems, aiding in the analysis of system behavior. In structural analysis, they provide insights into the stability and performance of structures.

Statistics

• Covariance Matrix: Eigenvalues are utilized in multivariate statistics to analyze covariance matrices, offering information about the spread and orientation of data.

Computer Graphics

• Principal Component Analysis (PCA): Eigenvalues are applied in PCA to identify the principal components of a dataset, reducing dimensionality while preserving critical information.

Control Systems

• System Stability: Eigenvalues of the system matrix are crucial for assessing the stability of a control system, ensuring that the system’s response remains bounded through stability analysis.

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