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 (P^T = P^-1) and D is a diagonal matrix. This is possible for all real symmetric matrices.

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 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, and solving systems of differential equations.

4. Using the Calculator

Tips: Enter the elements of your symmetric matrix as comma-separated values. For a 2×2 matrix, enter 4 values; for 3×3, enter 9 values, etc.

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, where P^-1 = P^T, while regular diagonalization doesn't have this requirement.

Q3: What are the applications of orthogonal diagonalization?
A: Applications include PCA in statistics, normal modes in physics, and solving systems of linear differential equations.

Q4: Why must the matrix be symmetric?
A: Only symmetric matrices are guaranteed to have real eigenvalues and a full set of orthogonal eigenvectors.

Q5: How accurate are the numerical results?
A: Accuracy depends on the implementation, but numerical methods may have small errors for ill-conditioned matrices.