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Diagonalization

Source lecture(s): PHY621 Lec2

Intuition

If a matrix has enough eigenvectors, rotate coordinates so the matrix becomes diagonal.

Formal Definition

\(A\) is diagonalizable if \(A=PDP^{-1}\) where \(D\) is diagonal and \(P\) contains eigenvectors as columns.

Mathematical Formulation

\[A=P\,\text{diag}(\lambda_1,\ldots,\lambda_N)\,P^{-1}\]

Derivation

Write \(AP=PD\). The \(j\)-th column gives \(A\vec{v}_j=\lambda_j\vec{v}_j\), which is exactly the eigenvalue equation.

Worked Example

A diagonal matrix is already in the form \(D\); its eigenvectors are the standard basis.

Common Mistakes

  • Assuming every matrix is diagonalizable.
  • Using the same eigenvector for repeated roots without checking degeneracy.

Quiz

Q1. What guarantees diagonalizability?

Answer

Distinct eigenvalues or a complete set of linearly independent eigenvectors.