Calculator
Tip: Fractions like 3/4 are accepted. Blank fields are treated as zero.
Formula used
A 4×4 matrix A is diagonalizable when there exists an invertible matrix P and a diagonal matrix D such that:
- D contains eigenvalues on the diagonal.
- Columns of P are the corresponding eigenvectors.
- Verification uses the maximum absolute entry of A − P D P-1.
For symmetric matrices, the eigenvectors are orthogonal, so P-1 = PT.
How to use this calculator
- Enter all 16 matrix values a11 through a44.
- Choose a solver mode, or keep Auto.
- Set tolerance and iterations for accuracy.
- Press Submit to see eigenvalues, P, D, and error.
- Use CSV or PDF buttons to export your result.
Example data table
| Example matrix A (block form) | Expected eigenvalues | Diagonalizable? |
|---|---|---|
| [[5,2,0,0],[2,5,0,0],[0,0,3,1],[0,0,1,3]] | 7, 3, 4, 2 | Yes (symmetric) |
| [[4,1,0,0],[0,3,0,0],[0,0,2,0],[0,0,0,1]] | 4, 3, 2, 1 | Yes (triangular with distinct diagonal) |
| [[1,1,0,0],[0,1,0,0],[0,0,2,1],[0,0,0,2]] | 1 (×2), 2 (×2) | Often no (defective Jordan blocks) |
Examples use simple structures so results are easy to verify by hand.
FAQs
1) What does diagonalization mean?
It rewrites a matrix as A = P D P-1, where D is diagonal. This makes powers of A easier, because Ak = P Dk P-1.
2) When is a 4×4 matrix diagonalizable?
It must have four linearly independent eigenvectors. Distinct eigenvalues guarantee this. Repeated eigenvalues can still work, but only if the eigenspaces provide enough independent vectors.
3) Why do I see “Not verified”?
The calculator checks how close P D P-1 is to your input. A large error can mean the matrix is not diagonalizable, or the numeric eigenvectors are unstable. Try symmetric mode for symmetric matrices.
4) Which mode should I use?
Use Auto first. If your matrix is symmetric, the symmetric solver is usually faster and more accurate. For general matrices, the QR method estimates eigenvalues, then uses inverse iteration for eigenvectors.
5) Can this handle complex eigenvalues?
This version targets real diagonalization and may struggle with matrices whose eigenvalues are complex. If you suspect complex pairs, consider transforming the matrix or using a tool that supports complex arithmetic explicitly.
6) What tolerance and iterations are reasonable?
Start with tolerance 1e-10 and 500 iterations. If results look noisy, increase iterations. If the matrix entries are very large or very small, scale the matrix to improve numerical stability.
7) Why are eigenvalues “approximate”?
Numerical algorithms use floating-point arithmetic, so tiny differences are expected. Rounding controls display only. The verification error is a better indicator of whether the computed diagonalization is trustworthy.
8) What do the CSV and PDF exports include?
They include your input matrix, estimated eigenvalues, and any computed matrices P, D, P-1, plus notes and the reconstruction error for quick sharing.