Linear Algebra made simple — Eigen Vectors and Eigen Values

Eigenvalues and eigenvectors are fundamental concepts in linear algebra that provide insight into the geometric properties of linear transformations.

Given a square matrix A, an eigenvector v is a non-zero vector that, when the matrix A is applied to it, only scales the vector by a constant factor, without changing its direction. This scaling factor is called the eigenvalue λ

Mathematically, this relationship is expressed as:

Av=λv

Where:

• A is an n×n matrix (the transformation matrix),
• v is the eigenvector,
• λ is the eigenvalue.

In simple terms,

• Eigenvector: A vector whose direction remains unchanged by the transformation.
• Eigenvalue: A scalar that indicates how much the eigenvector is stretched or compressed during the transformation.

Let us understand with example,

Consider the matrix A

Our goal is to find the eigen values and eigen vectors of this matrix.

To find the eigenvalues, we solve the characteristic equation, which is

For our Matrix A the above equation looks like this

Now, we calculate the determinant

Solving this equation we get 2 values for λ which is λ = 5 , λ = 2

These are called eigen values

From this, let us find the eigen vectors

For λ = 5, the eigen vector is

(A — 5I)v = 0

Now we can solve

The equations are

The equations are redundant and leads to the same value as

This will be satisfied for any values we can take the simplest form that satisfies this equation as 1 and 1

Eigen Vector for λ = 2 is

From the above equations we get

So the eigen vector is any scalar multiple of 1 and -2 choosing the simplest form

Summary would be —

• For λ=5, the corresponding eigenvector means that any vector along the direction (1,1) is scaled by 5 when transformed by the matrix A
• For λ=2, the corresponding eigenvector means that any vector along the direction (1,-2) is scaled by 2 when transformed by the matrix A

Here is the graph representing the eigenvectors and the transformation of the matrix A.

• The red arrow represents the first eigenvector corresponding to eigenvalue λ=5.
• The blue arrow represents the second eigenvector corresponding to eigenvalue λ=2.
• The green and purple arrows show how the standard unit vectors (x-axis and y-axis) are transformed by the matrix A.

The eigenvectors show the directions in which the transformation acts purely as scaling, without changing direction, while the transformed unit vectors illustrate how the matrix A distorts the space.