Eigenvectors

The eigenvectors are the non-vectors that do not change direction when a linear transformation is applied to them. In linear algebra, an eigenvector helps in complex transformations of a matrix. For a square matrix A and a vector v, the eigenvector is represented as:

Av = λv, where λ is the eigenvalue. 
 

In linear algebra, the eigenvector and eigenvalues are fundamental concepts. Eigenvectors are the non-zero vectors that satisfy the equation Av = λv. Where A is the square matrix, v is the vector, and λ is the eigenvalue. The eigenvalue is the scalar λ associated with the eigenvector. 
Geometrically, eigenvectors are used to find the directions that remain unchanged after transformation. For example, in a rotation matrix, the eigenvector lies along the axis of rotation. Algebraically, an eigenvector satisfies the equation (A - λI)v = 0, where I is the identity matrix. 
 

Eigenvectors are non-zero vectors when transformed into matrix, it does not change direction. It only changes the length. In this section, we will discuss the difference between the eigenvector and other vectors.  
 

Eigenvectors and eigenvalues are used to simplify the matrix operations and to understand the linear transformation. In this section, let’s learn how to find eigenvectors and eigenvalues. 

Step 1: Finding the characteristic equation
To find the eigenvalue λ of a square matrix A, we use the formula det(A - λI) = 0.

Step 2: Substitute eigenvalues
After finding the value of λ, solve (A - λI)v = 0 and find the value of v. 

Step 3: Normalize the eigenvector
If unit vectors are required, we divide each eigenvector by its magnitude to normalize. 

For example, let’s find the eigenvalues and eigenvectors of the matrix A = 1221
Finding the value of λ, 
det(A - λI) = 12 - λ 2 - λ1
= (2 - λ)2 - 1
= λ2 - 2λ + 4 - 1
= λ2 - 2λ + 3
Solving λ2 - 2λ + 3 = 0
Using the quadratic equation to find the value of λ
λ = -b ± b2 - 4ac2a
Substituting the value:
λ = -(-4) ± (-4)2 - 4 × 1 × 32 × 1
λ = 4 ± 16 - 122 
λ = 4 ± 22 
λ = 4 + 22                  λ = 4 - 22 
λ = 3, λ = 1

Finding the eigenvector using the equation (A - λI)v
For λ = 1:
(A - 1I)v = 1221 - 01 10 = 1111 

 1111 yx = 0
x + y = 0
y = -x

So, any vector of the form is:
v = -x   x   is an eigenvector

If x = 1 v becomes
v1 = -1   1

For λ = 3:
(A - 3I)v = 1221 - 03 30 =     1-1-1   1
    1-1-1   1 yx = 0

-x + y = 0
y = x

So, v2 = xx
If x = 1, v2 become
v2 = 11
So, the eigenvector corresponding to the eigenvalues λ = 1 and λ = 3 are:
λ = 1 → -1   1
λ = 2 → 11
 

The eigenvectors and eigenvalues follow many properties that help to understand the matrix behavior. In this section, we will learn some properties of eigenvectors and eigenvalues. 

• The eigenvectors associated with the distinct eigenvalues are linearly independent. 

• An n × n matrix can have up to n eigenvalues, including repeated values known as repeated or multiple eigenvalues. 

• For any symmetric matrix, the eigenvalues are real, and the eigenvectors are mutually perpendicular (orthogonal)

• The sum of the eigenvalues is equal to the total of its diagonal elements, trace. The product of the eigenvalues is equal to the determinant of the matrix. 
   

Based on the type of matrices, the eigenvalues and eigenvectors follow a specific pattern. Let’s understand some types of matrices and their eigenvectors and eigenvalues. 

• Symmetric Matrices: All symmetric matrices have real eigenvalues and orthogonal eigenvectors.

• Diagonal Matrices: For a diagonal matrix, the eigenvectors are the standard basic vectors, and the eigenvalues are the entries along the diagonal

• Singular Matrices: In these types of matrices, at least one eigenvalue is zero. 
   

In real life, eigenvectors are used in fields like principal component analysis, physics, engineering, and computer graphics. In this section, we will discuss some real-world applications of eigenvectors. 

• In principal component analysis, we use eigenvectors to transform data to a new coordinate system. For example, reducing dimensionality in image processing

• In physics, to determine the possible measurement outcomes and quantum states, we use eigenvectors. 

• In engineering, eigenvalues are used for stability analysis in control systems and structural engineering. For example, they help determine whether a system remains stable under various conditions. 

• Eigenvectors are used in computer graphics for transformations, animations, image compression, and shape analysis. For example, rotation matrices in 3D modeling.