8) Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors

1. What are Eigenvalues and Eigenvectors?

When a linear transformation is applied to a vector, some vectors only scale without changing direction. These special vectors are called eigenvectors, and the scaling factor is called the eigenvalue.

Mathematical Definition

A v = λ v

Where A is a square matrix, v is an eigenvector, and λ is the eigenvalue.

Real-World Meaning

  • Physics: Eigenvectors represent stable states (e.g., vibration modes in mechanics).
  • Machine Learning: PCA finds important patterns using eigenvectors.
  • Computer Graphics: Used in transformations like shearing and rotations.

2. How to Compute Eigenvalues and Eigenvectors

To find the eigenvalues of a matrix A, solve the characteristic equation:

det(A - λI) = 0

Steps:

  1. Compute A - λI, where I is the identity matrix.
  2. Solve det(A - λI) = 0 to find eigenvalues λ.
  3. For each λ, solve (A - λI)v = 0 to find eigenvectors.

Example

For matrix:

A = [ [4, 2], [1, 3] ]

Solving det(A - λI) = 0 gives eigenvalues, which are then used to find eigenvectors.

3. Eigenvalues in Transformations

Eigenvalues determine how transformations scale space:

  • λ > 1: Stretching.
  • 0 < λ < 1: Shrinking.
  • λ = -1: Reflection.
  • λ = 0: Collapsing to a lower dimension.

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