1. What is Orthogonal Diagonalization?

Orthogonal diagonalization is the process of decomposing a symmetric matrix A into the form A = PDPᵀ, where P is an orthogonal matrix (Pᵀ = P⁻¹) and D is a diagonal matrix containing the eigenvalues of A.

2. How Does the Calculator Work?

The calculator performs the following steps:

1. Verifies the input matrix is symmetric (A = Aᵀ)
2. Computes the eigenvalues of the matrix
3. Finds corresponding orthogonal eigenvectors
4. Constructs P from the eigenvectors and D from the eigenvalues

3. Importance of Orthogonal Diagonalization

Applications: Used in principal component analysis (PCA), quadratic forms, systems of differential equations, and many areas of physics and engineering.

4. Using the Calculator

Instructions: Enter a symmetric 3×3 matrix (where aᵢⱼ = aⱼᵢ for all i,j). The calculator will verify symmetry and compute the orthogonal diagonalization if 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 are eigenvalues and eigenvectors related to this?
A: D contains eigenvalues of A, and P's columns are corresponding orthonormal eigenvectors.

Q3: What's the difference between diagonalization and orthogonal diagonalization?
A: Orthogonal diagonalization requires P to be orthogonal (P⁻¹ = Pᵀ), which is only possible for symmetric matrices.

Q4: Why is orthogonal diagonalization useful?
A: It simplifies matrix computations and reveals important geometric properties of the linear transformation.

Q5: Can non-square matrices be orthogonally diagonalized?
A: No, orthogonal diagonalization only applies to square matrices that are symmetric.