Calculator
Enter a real square matrix. Orthogonal diagonalization requires symmetry.
Formula Used
A real symmetric matrix can be written as:
A = QDQᵀ
Here, Q is an orthogonal matrix. Its columns are normalized eigenvectors. Also, D is diagonal. Its diagonal entries are eigenvalues.
The orthogonality condition is:
QᵀQ = I
The calculator also verifies:
QᵀAQ = D and A ≈ QDQᵀ
For a real matrix, orthogonal diagonalization is possible when:
A = Aᵀ
How to Use This Calculator
- Select the matrix size from 2 × 2 to 5 × 5.
- Enter each matrix value. Fractions like
3/4are accepted. - Choose decimal places, tolerance, sorting, and iteration limit.
- Keep symmetric-part mode off for exact orthogonal diagonalization.
- Press the calculate button.
- Review eigenvalues, Q, D, QᵀAQ, and the residual error.
- Use CSV or PDF buttons to save the result.
Example Data Table
| Matrix A | Symmetric? | Eigenvalues | One valid diagonal form |
|---|---|---|---|
[[2, 1, 0], [1, 2, 0], [0, 0, 3]]
|
Yes | 1, 3, 3 | D = diag(1, 3, 3) |
[[4, 1], [1, 4]]
|
Yes | 5, 3 | D = diag(5, 3) |
[[3, 2], [1, 3]]
|
No | Not valid directly | Use only after symmetrizing, if needed. |
Orthogonal Diagonalization Guide
What It Means
Orthogonal diagonalization is a key idea in linear algebra. It rewrites a suitable square matrix as A = QDQᵀ. The matrix Q contains orthonormal eigenvectors. The matrix D contains eigenvalues on its diagonal. This form keeps lengths and angles stable.
Why Symmetry Matters
A real matrix can be orthogonally diagonalized when it is symmetric. That means A equals Aᵀ. Symmetry makes the eigenvectors from different eigenvalues perpendicular. The calculator checks this condition first. It also measures the largest mismatch between mirrored entries.
Jacobi Rotation Method
The tool uses the Jacobi rotation method. This method repeatedly removes the largest off diagonal value. Each rotation is orthogonal. So the accumulated vector matrix stays stable. The process stops when off diagonal values are smaller than the selected tolerance.
Reading the Result
Results include eigenvalues, eigenvectors, diagonal matrix D, and matrix Q. You also get QᵀAQ and QDQᵀ. These two checks help confirm the answer. A small residual means the diagonalization is accurate.
Validation Checks
Trace and determinant checks are also useful. The trace equals the sum of eigenvalues. The determinant equals their product. These values provide quick validation. They help catch typing mistakes before using the result.
Advanced Options
The calculator supports several options. You can choose matrix size, decimal places, tolerance, sorting, and iteration limits. You can also set a zero threshold. Small values below this level are printed as zero. This makes reports easier to read.
Graph Meaning
Graphs show the eigenvalue spread. The bar chart helps compare dominant directions. Large positive values stretch space. Negative values flip a direction and scale it. Near zero values may indicate a flat or weak direction.
Common Uses
Orthogonal diagonalization is common in quadratic forms. It is also used in statistics, engineering, physics, and computer graphics. In each case, the method finds independent directions. The original matrix becomes easier to understand.
Teaching and Reports
For teaching, the calculator shows matrices in a clean layout. Students can compare the input with the diagonal form. Teachers can export examples for notes. Engineers can inspect residuals before using a model. The workflow keeps every important number visible.
Best Practice
Use exact entries when possible. Avoid rounding too early. Check the symmetry warning. Then review the reconstruction error. The best result has orthonormal columns, a diagonal D, and a tiny residual.
FAQs
1. What is orthogonal diagonalization?
It is the process of writing a matrix as A = QDQᵀ, where Q is orthogonal and D is diagonal. The columns of Q are orthonormal eigenvectors.
2. Which matrices can be orthogonally diagonalized?
Every real symmetric matrix can be orthogonally diagonalized. If a real matrix is not symmetric, this calculator warns you before continuing.
3. What does Q represent?
Q is the orthogonal eigenvector matrix. Each column is a normalized eigenvector. Its columns should satisfy QᵀQ = I.
4. What does D represent?
D is the diagonal matrix of eigenvalues. Each diagonal value matches the eigenvector in the same column position of Q.
5. Why is the residual important?
The residual measures how closely QDQᵀ reconstructs the matrix. A smaller value means the diagonalization is more accurate.
6. Can I enter fractions?
Yes. You can enter values such as 1/2, -3/4, or 5/2. The calculator converts them into decimal values internally.
7. Why are repeated eigenvalues sometimes different in Q?
Repeated eigenvalues can have many valid orthonormal eigenvector bases. Your Q may differ from a textbook but still be correct.
8. What does symmetric-part mode do?
It replaces A with (A + Aᵀ) / 2. This does not diagonalize the original non-symmetric matrix. It analyzes its nearest symmetric part.