Eigenvalue and Diagonalisation Calculator
Find eigenvalues and eigenvectors with their multiplicities, diagonalise a matrix (P⁻¹AP = D), apply Cayley–Hamilton, raise A to a power, and reduce a quadratic form to canonical form, all with exact fractions.
About the Eigenvalue and Diagonalisation Calculator
An eigenvector X of A keeps its direction when A acts on it: AX = λX. The eigenvalues λ are the roots of the characteristic equation |A − λI| = 0, which for a 3 × 3 matrix is λ³ − S₁λ² + S₂λ − S₃ = 0, where S₁ is the trace, S₂ the sum of the principal minors and S₃ the determinant.
A is diagonalisable when it has n independent eigenvectors: put them in the columns of P and P⁻¹AP is the diagonal matrix D of eigenvalues, which makes powers easy: Aᵏ = PDᵏP⁻¹. A real symmetric matrix is always diagonalisable by an orthogonal P, which turns the quadratic form XᵀAX into the sum of squares λ₁y₁² + λ₂y₂² + λ₃y₃² (orthogonal reduction).
The formula
|A − λI| = 0; P⁻¹AP = D; Aᵏ = PDᵏP⁻¹
- Cayley–Hamilton
- every square matrix satisfies its own characteristic equation, so A⁻¹ = (A² − S₁A + S₂I) ÷ S₃ for 3 × 3
- Index, signature
- the number of positive eigenvalues; positives minus negatives
Worked example
A = [[1, 1, 3], [1, 5, 1], [3, 1, 1]]
- λ³ − 7λ² + 36 = 0
- λ = −2, 3, 6 with X = (1, 0, −1), (1, −1, 1), (1, 2, 1)
- Q = x² + 5y² + z² + 2xy + 2yz + 6zx is indefinite (rank 3, index 2, signature 1)
Questions people ask
When is a matrix not diagonalisable?
When a repeated eigenvalue has fewer independent eigenvectors than the number of times it repeats, as in [[1, 1], [0, 1]].
Do I need different eigenvalues?
No. Distinct eigenvalues guarantee diagonalisability, but a repeated eigenvalue can still give enough eigenvectors (for example the identity matrix).
How does Cayley–Hamilton give the inverse?
Multiply A³ − S₁A² + S₂A − S₃I = O by A⁻¹ and rearrange: A⁻¹ = (A² − S₁A + S₂I) ÷ S₃.