Math Explained to Programmers: From Eigenvectors to Eigen Decomposition

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Eigenvalues and Eigenvectors

Given an n × n matrix A, if there exists a non-zero vector v and a scalar λ, such that:

Av = λv

Then:

• λ is called an eigenvalue of A.
• v is the eigenvector corresponding to eigenvalue λ.

Characteristic Equation

Rewrite the equation as:

(A - λI)v = 0

For v ≠ 0, the following must hold:

det(A - λI) = 0

This equation is called the characteristic polynomial.

Types of Eigenvalues and Eigenvectors

Case 1: Distinct Real Eigenvalues

• Each eigenvalue has its independent eigenvector.
• Eigenvectors are linearly independent.

Case 2: Repeated Real Eigenvalues

• If geometric multiplicity (number of independent eigenvectors) equals algebraic multiplicity (multiplicity of the eigenvalue), the matrix is diagonalizable.
• If geometric multiplicity is less than algebraic multiplicity, the matrix is not diagonalizable.

Case 3: Complex Eigenvalues

• Occur in conjugate pairs (a + bi, a —…