Formula used
Diagonalization rewrites a square matrix using eigenvectors and eigenvalues. The columns of P are independent eigenvectors. Matching eigenvalues appear along the diagonal of D.
The characteristic equation is det(λI − A) = 0. Each root is an eigenvalue. The nullspace of A − λI gives its eigenspace.
How to use this calculator
Enter all nine matrix entries. Choose real or complex diagonalization. Select precision, normalization, ordering, and optional matrix functions.
Press the calculation button. Review the status first. Then inspect eigenvalues, multiplicities, eigenspace bases, P, D, and verification residuals.
Use parameters for reusable examples. Write entries like a+1 or sqrt(2). Then assign values using a=3 or another valid expression.
What diagonalization means
A diagonal matrix is easier to use than a general matrix. Its powers only require powering diagonal entries. This simplifies recurrences, differential systems, Markov models, and repeated transformations.
A matrix is diagonalizable when it has enough independent eigenvectors. A 3×3 matrix needs three independent eigenvectors. Distinct eigenvalues usually provide this requirement automatically.
Algebraic and geometric multiplicity
Algebraic multiplicity counts an eigenvalue's repetitions in the characteristic polynomial. Geometric multiplicity counts independent eigenvectors in its eigenspace. Geometric multiplicity cannot exceed algebraic multiplicity.
A repeated eigenvalue does not automatically prevent diagonalization. It becomes a problem when its eigenspace is too small. The calculator reports both multiplicities clearly.
Real and complex diagonalization
A real matrix may have complex eigenvalues. Such a matrix may fail over real numbers but succeed over complex numbers. Select complex mode to allow complex diagonal entries and eigenvectors.
Complex results can appear in rectangular or polar form. Rectangular form is usually better for algebra. Polar form can help interpret magnitude and phase.
Orthogonal diagonalization
Every real symmetric matrix is orthogonally diagonalizable. Its eigenvectors can form an orthonormal basis. Then P inverse equals P transpose, improving stability and interpretation.
The calculator detects real symmetry automatically. It can normalize and orthogonalize eigenspace bases. Verification then uses QDQ transpose.
Why verification matters
Numerical roots and eigenvectors contain rounding error. The reconstructed matrix may differ slightly from the input. Residual matrices reveal the size and location of these differences.
Small residual values support the result. Large values may indicate unstable eigenvectors, poor scaling, or near-repeated roots. Adjust precision and tolerances carefully.
Jordan form for defective matrices
A defective matrix lacks enough independent eigenvectors. It cannot be diagonalized normally. Jordan form replaces a fully diagonal matrix with diagonal blocks containing selected superdiagonal ones.
This page provides a structural Jordan suggestion. Exact Jordan chains can be sensitive numerically. Symbolic computer algebra remains preferable for exact defective matrices.
Matrix powers and functions
When A equals PDP inverse, A to the nth power equals P times D to the nth power times P inverse. Large powers become much easier.
The same idea applies to analytic matrix functions. Exponentials use e to each diagonal entry. Principal square roots use the selected complex square-root branch.
Example data
| Matrix type | Example | Expected outcome |
|---|---|---|
| Diagonal | diag(2,3,5) | Already diagonal |
| Distinct real eigenvalues | Upper triangular with 1,2,4 | Usually diagonalizable |
| Symmetric | [[2,1,0],[1,2,0],[0,0,3]] | Orthogonally diagonalizable |
| Defective | [[2,1,0],[0,2,0],[0,0,3]] | Not diagonalizable |
| Complex spectrum | Rotation block plus one real value | Requires complex mode |
| Scalar matrix | 4I | Every nonzero vector is an eigenvector |
Frequently asked questions
What matrices can be diagonalized?
A 3×3 matrix is diagonalizable over a chosen field when it has three linearly independent eigenvectors in that field.
Do three distinct eigenvalues guarantee diagonalization?
Yes. Three distinct eigenvalues provide three independent eigenvectors over a field containing those eigenvalues.
Can a repeated eigenvalue still work?
Yes. Its eigenspace must contain enough independent vectors to match its algebraic multiplicity.
Why does real mode reject some matrices?
A real matrix can possess nonreal eigenvalues. Real diagonalization then fails, although complex diagonalization may remain possible.
What is a defective matrix?
A defective matrix lacks a complete eigenvector basis. Its total geometric multiplicity is smaller than three.
Why can eigenvectors look different elsewhere?
Any nonzero scalar multiple of an eigenvector is also valid. Different bases can represent the same eigenspace.
What does the residual show?
It displays A minus PDP inverse. Values near zero indicate successful numerical reconstruction.
Is rationalized display exact?
It recognizes nearby simple fractions. General cubic roots remain numerical unless they naturally simplify within tolerance.