Diagonalization Result
The result appears here after pressing the submit button.
Eigenvalue Chart
Eigenpairs
| # | Eigenvalue | Normalized Eigenvector |
|---|
Original Matrix A
Orthogonal Matrix P
Diagonal Matrix D
Reconstructed Matrix PDPᵀ
Verification Notes
Calculator Inputs
Enter a real symmetric matrix. Orthogonal diagonalization is guaranteed for real symmetric matrices. The calculator uses an iterative Jacobi method and reports accuracy checks.
Example Data Table
Use these examples to test the calculator quickly.
| Example | Matrix | Expected behavior | Action |
|---|---|---|---|
| Simple 2 × 2 | [[2, 1], [1, 2]] | Two orthogonal eigenvectors and two real eigenvalues. | |
| Symmetric 3 × 3 | [[4, 1, 1], [1, 3, 0], [1, 0, 2]] | Three real eigenvalues and an orthonormal basis. | |
| Repeated eigenvalue | [[3, 0, 0], [0, 3, 1], [0, 1, 3]] | Orthogonal diagonalization still works with repeated roots. | |
| Symmetric 4 × 4 | [[6, 2, 0, 1], [2, 5, 1, 0], [0, 1, 4, 2], [1, 0, 2, 3]] | Useful for checking larger matrix accuracy. |
Formula Used
Orthogonal diagonalization:
A = PDPᵀ
PᵀP = I
D = diag(λ₁, λ₂, ..., λₙ)
Avᵢ = λᵢvᵢ
Residual error = ||A − PDPᵀ||F
Orthogonality error = ||PᵀP − I||F
The columns of P are normalized eigenvectors. The diagonal entries of D are eigenvalues. For a real symmetric matrix, the spectral theorem guarantees this decomposition.
How to Use This Calculator
- Select the matrix size from 2 × 2 to 6 × 6.
- Enter all values in the matrix grid.
- Keep the matrix symmetric, or choose the averaging option.
- Set tolerance, iteration limit, sorting, and decimals.
- Press the submit button to compute P, D, and PDPᵀ.
- Review the residual and orthogonality errors.
- Use the CSV or PDF buttons to export the result.
Orthogonal Diagonalization Guide
What It Means
Orthogonal diagonalization rewrites a square matrix in a simpler form. It works best for real symmetric matrices. The matrix A becomes PDPᵀ. The matrix D contains eigenvalues. The matrix P contains unit eigenvectors. Since P is orthogonal, its inverse equals its transpose.
Why Symmetry Matters
A real symmetric matrix has matching entries across the main diagonal. This special structure gives real eigenvalues. It also gives perpendicular eigenvectors. These facts make the decomposition stable and meaningful. Non-symmetric matrices may not have an orthogonal diagonalization.
How the Method Works
This calculator uses Jacobi rotations. Each rotation reduces an off-diagonal value. Repeated rotations push the matrix toward diagonal form. At the same time, the calculator updates the eigenvector matrix. The final diagonal entries become eigenvalues.
Reading the Output
The table lists each eigenvalue and its normalized eigenvector. The matrix P places those eigenvectors as columns. The matrix D places eigenvalues on the main diagonal. The reconstructed matrix PDPᵀ should match the original matrix.
Accuracy Checks
Residual error measures reconstruction quality. A smaller residual means a closer match. Orthogonality error checks whether PᵀP is close to I. A small value confirms that the eigenvectors are nearly orthonormal.
Practical Uses
Orthogonal diagonalization appears in quadratic forms, principal axes, statistics, vibrations, data analysis, and numerical linear algebra. It helps reveal independent directions in a system. It also turns many matrix operations into easier diagonal operations.
Study Tip
Always check symmetry before diagonalizing. Then compare A and PDPᵀ. If the errors are small, the result is reliable. If the errors are large, use stricter tolerance or review the input values.
FAQs
1. What is orthogonal diagonalization?
It writes a matrix as A = PDPᵀ, where P is orthogonal and D is diagonal. This form separates the matrix into eigenvectors and eigenvalues.
2. Which matrices can be orthogonally diagonalized?
Every real symmetric matrix can be orthogonally diagonalized. This is guaranteed by the spectral theorem. Non-symmetric matrices may fail this condition.
3. What does matrix P represent?
Matrix P contains normalized eigenvectors as columns. These vectors are mutually perpendicular when the input matrix is real and symmetric.
4. What does matrix D represent?
Matrix D is diagonal. Its diagonal entries are the eigenvalues of the original matrix, ordered according to the selected sorting option.
5. Why is the residual error important?
The residual error measures how closely PDPᵀ reconstructs A. A very small value means the computed diagonalization is accurate.
6. Can repeated eigenvalues be handled?
Yes. Symmetric matrices with repeated eigenvalues still have an orthonormal eigenbasis. The calculator can produce one valid orthogonal basis.
7. Why does the calculator reject some matrices?
Orthogonal diagonalization requires symmetry. If the matrix is not symmetric, choose the averaging option or correct the mismatched entries.
8. What does the eigenvalue chart show?
The chart displays eigenvalues as bars. It helps compare sign, size, and spread across the diagonalized matrix.