Calculator
Enter a 2×2 or 3×3 matrix. The tool estimates eigenvalues, eigenvectors, P and D matrices, reconstruction accuracy, and exportable outputs.
Example data table
| Example | Matrix | Typical outcome | Statistics use |
|---|---|---|---|
| Symmetric 2×2 covariance sample | [[4, 1], [1, 3]] | Real eigenvalues and stable P, D decomposition | Small variance-direction analysis |
| Symmetric 3×3 correlation style sample | [[1, 0.4, 0.2], [0.4, 1, 0.3], [0.2, 0.3, 1]] | Orthogonal-style eigenvectors and real diagonal entries | PCA teaching and feature relationships |
| Repeated-eigenvalue stress sample | [[2, 1], [0, 2]] | May fail full diagonalization | Shows when P and D form is unavailable |
Formula used
Main decomposition: A = P D P-1
Here, A is the original matrix, P stores eigenvectors as columns, and D is a diagonal matrix of eigenvalues.
Eigen relation: A vi = λi vi
Each eigenvector vi keeps its direction under transformation. Its scale change equals eigenvalue λi.
Diagonal matrix: D = diag(λ1, λ2, ..., λn)
The diagonal entries summarize scaling along principal directions. This is useful when studying covariance, correlation, and repeated linear transformations.
Useful checks: trace(A) = Σ λi and det(A) = Π λi
The calculator also reports trace, determinant, spectral radius, symmetry, and reconstruction error to support interpretation.
How to use this calculator
- Choose a 2×2 or 3×3 matrix size.
- Enter the matrix values in the form fields.
- Set your preferred decimal precision and graph mode.
- Click Calculate P and D to generate outputs.
- Review eigenvalues, eigenvectors, determinant, trace, and symmetry.
- Inspect the P matrix, D matrix, and reconstruction section.
- Use the Plotly graph to compare eigenvalue components.
- Export the result as CSV or PDF for documentation.
Frequently asked questions
1) What does the P matrix represent?
P stores eigenvectors as columns. Each column shows a direction that keeps its orientation under the matrix transformation while only its scale changes.
2) What does the D matrix represent?
D places eigenvalues on the diagonal. Each diagonal entry measures how strongly the original matrix scales the matching eigenvector from P.
3) Why is diagonalization useful in statistics?
It helps interpret covariance or correlation structure, simplify repeated matrix operations, and support PCA-style reasoning about independent directions of variation.
4) Does every matrix have a valid P and D decomposition?
No. A matrix needs enough linearly independent eigenvectors. When that condition fails, the calculator flags the matrix as not fully diagonalizable.
5) What happens if the eigenvalues are complex?
The tool still reports them. Complex eigenvalues usually appear in non-symmetric matrices, while symmetric covariance-style matrices commonly give real eigenvalues.
6) Why does the calculator check reconstruction error?
The reconstruction test verifies that P D P-1 recreates the original matrix within rounding tolerance. Smaller error means a cleaner numerical result.
7) Can I use a 3×3 covariance matrix here?
Yes. Enter a 3×3 symmetric matrix to inspect eigenvalues, eigenvectors, and decomposition. This is helpful for compact PCA-style examples.
8) What do the CSV and PDF exports include?
They include summary metrics, eigenvalues, the original matrix, P matrix, D matrix, and the reconstructed matrix when available.