1. What is Orthogonal Diagonalization?

Orthogonal diagonalization is the process of decomposing a symmetric matrix A into the form \( A = PDP^T \), where P is an orthogonal matrix (its columns are orthonormal eigenvectors of A) and D is a diagonal matrix containing the eigenvalues of A.

2. How Does the Calculator Work?

The calculator performs the following steps:

Where:

• \( A \) — Input symmetric matrix
• \( P \) — Orthogonal matrix of eigenvectors
• \( D \) — Diagonal matrix of eigenvalues
• \( P^T \) — Transpose of P

Explanation: The calculator first verifies the matrix is symmetric, then computes eigenvalues and corresponding orthonormal eigenvectors to construct P and D.

3. Importance of Orthogonal Diagonalization

Details: Orthogonal diagonalization is crucial in many applications including principal component analysis (PCA), quadratic forms simplification, and solving systems of differential equations.

4. Using the Calculator

Tips: Enter a symmetric matrix with elements separated by commas or spaces. The matrix must be square and symmetric (A = A^T) for orthogonal diagonalization to be possible.

5. Frequently Asked Questions (FAQ)

Q1: What matrices can be orthogonally diagonalized?
A: Only symmetric (or Hermitian in complex case) matrices can be orthogonally diagonalized.

Q2: How is this different from regular diagonalization?
A: Orthogonal diagonalization uses an orthogonal matrix P (P^T = P^{-1}), while regular diagonalization may use any invertible matrix.

Q3: What if my matrix isn't symmetric?
A: Non-symmetric matrices may still be diagonalizable, but not via an orthogonal matrix. They require Jordan form or singular value decomposition.

Q4: Are there numerical precision limitations?
A: Yes, eigenvalue computation can be sensitive to numerical errors, especially for nearly singular matrices.

Q5: What applications use orthogonal diagonalization?
A: PCA in statistics, normal modes in physics, and many engineering applications involving quadratic forms.