Eigenvalue of a matrix

Eigenvalues and eigenvectors are essential concepts in linear algebra.Eigenvalue of a matrix. Here’s an explanation:

Eigenvalues and Eigenvectors:

1. Eigenvalue (λ): A scalar representing how much an eigenvector is stretched or compressed during a linear transformation.

2. Eigenvector (v): A non-zero vector applied in a linear transformation just gets scaled (not rotated).

Mathematical Definition

If A is a square matrix and $v$ is a non-zero vector, then $v$ is an eigenvector of $A$ with eigenvalue λ if:

• $A$: The square matrix (linear transformation)
• $v$: The eigenvector.
• $\lambda$: The eigenvalue.

Finding Eigenvalues and Eigenvectors

1. Find the Eigenvalues($\lambda$):

• Rearrange the equation: Av-$\lambda v=0$
• Rewrite it as : (A−λI)v=0, where $I$ is the identity matrix.
• In the case of a non-trivial solution (v  is not equal to 0), the determinant of (A−λI) must be zero.
• Solve this characteristic equation to find λ.

2. Find the Eigenvectors(v):

• Substitute each 𝜆 back into  (A−λI)v=0.
• Solve the resulting system of linear equations to find v.

Example:

Step 1: Find Eigenvalues

The eigenvalue equation is:

where the identity matrix is denoted by I.

The determinant is

Simplify

Factorize

So, the eigenvalues are

Step 2: Find Eigenvectors

For each eigenvalue, solve (A−λI)v=0

Eigenvector for λ_1=3

Solve (A-3I)v=0

From the first row

Eigen vector

Eigenvector for λ_2=1

Solve (A-I)v=0

From the first row

Eigenvector

Final answer

• Eigenvalues: λ_1=3, λ_2=1
• Eigenvectors:

Why are Eigenvalues and Eigenvectors Important?

1. Understanding Transformations: Eigenvectors show the directions in which a transformation extends or compresses a space, whereas eigenvalues indicate how much stretching or compression occurs.

2. Data Reduction: Principal Component Analysis (PCA) is a technique that uses eigenvalues and eigenvectors to minimize data dimensionality while maintaining as much variability as possible.

3. Stability Analysis: Eigenvalues of a matrix are used in physics and engineering to examine system stability, such as whether a bridge can sustain particular forces or if a rocket’s trajectory is stable.

4.Graph Theory: The eigenvalues of adjacency matrices in graphs are studied for connectivity, clustering, and other features.