Diagonalisation Calculator

Work out diagonalisation instantly with clear inputs, formula shown and shareable results.

About Diagonalisation Calculator

A matrix is diagonalisable when its eigenvectors span the space, giving A = PDP⁻¹ with the eigenvalues on the diagonal of D. Distinct eigenvalues always guarantee this; a repeated eigenvalue may not.

Formula

Diagonalisation

A = PDP⁻¹, where the columns of P are eigenvectors and D holds the eigenvalues

Worked example

With the default values below, Diagonalisation Calculator returns Diagonalisability of Diagonalisable: A = PDP⁻¹ with D = diag(6, 1). Change any input above and every figure here is recalculated in your browser.

Inputs used

Entry a: 5
Entry b: 4
Entry c: 1
Entry d: 2

Results

Diagonalisability: Diagonalisable: A = PDP⁻¹ with D = diag(6, 1)
First eigenvalue: 6.00000000
Second eigenvalue: 1.00000000
Columns of P: (4, 1) and (4, -4)

Step by step

1Values used
2Diagonalisation
3Diagonalisability= Diagonalisable: A = PDP⁻¹ with D = diag(6, 1)
4First eigenvalue= 6.00000000
5Second eigenvalue= 1.00000000
6Columns of P= (4, 1) and (4, -4)

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For [[5,4],[1,2]]?

The trace is 7 and determinant 6, so the eigenvalues are 6 and 1 — distinct, hence diagonalisable.

Why is diagonalisation useful?

Powers become trivial: Aᵏ = PDᵏP⁻¹, and Dᵏ just raises each diagonal entry to the k.