Eigenvalue calculator

An eigenvalue calculator finds the numbers λ and the vectors v for which a matrix A just stretches v: Av = λv. Our Matrix Calculator does this for matrices from 2 by 2 up to 5 by 5. Choose Eigenvalues, type the matrix, and it gives the characteristic polynomial, the eigenvalues exactly where possible, and an eigenvector for each, with every step.

Eigenvalues are one of the Matrix Calculator's operations, so the first step is to switch to it. Fractions work in the entries, and the answers stay exact, including surds and complex eigenvalues.

Try it

Matrix Calculator

Multiply, invert and find determinants and eigenvalues

Open the calculatorGo to its page

How to use the eigenvalue calculator

1. In the Matrices section, choose Eigenvalues as the operation.
2. Set the size of matrix A, from 2 by 2 up to 5 by 5. It must be square.
3. Type the entries. Fractions such as 1/2 and negatives both work.
4. Read the eigenvalues, then the readouts: an eigenvector for each eigenvalue and the characteristic polynomial.
5. Follow the working: the characteristic equation, the polynomial, its factors, and the system solved for each eigenvector.

For a 2 by 2 matrix, the 2×2 transformation operation draws what the matrix does to a grid, which shows the eigenvector directions as lines that stay put. The Examples list loads an eigenvalue example to start from. For practice, Practice and help sets random matrix questions with worked solutions.

How eigenvalues are found

An eigenvalue λ makes A − λI singular, so its determinant is zero. That gives the characteristic equation:

Expanding the determinant gives a polynomial in λ, the characteristic polynomial, and its roots are the eigenvalues. For each eigenvalue, solve (A − λI)v = 0 to get an eigenvector v. For a 2 by 2 matrix there is a shortcut:

Worked examples you can check in the calculator

Example 1: a 2 by 2 matrix

Find the eigenvalues and eigenvectors of A with rows (5, 2) and (2, 2). The trace is 5 + 2 = 7 and the determinant is 5 × 2 − 2 × 2 = 6, so:

The eigenvalues are λ = 1 and λ = 6. For λ = 6, (A − 6I)v = 0 gives −v₁ + 2v₂ = 0, so v = (2, 1). For λ = 1, it gives 4v₁ + 2v₂ = 0, so v = (1, −2). Type the matrix with Eigenvalues chosen and the calculator gives these eigenvalues and eigenvectors.

Example 2: what an eigenvector looks like

For A with rows (4, 1) and (2, 3), the calculator finds λ = 2 and λ = 5, with eigenvectors (1, −2) and (1, 1). Multiplying out, A(1, 1) = (5, 5), five times (1, 1), and A(1, −2) = (2, −4), twice (1, −2). The matrix stretches each eigenvector along its own line:

Example 3: a 3 by 3 matrix

For a triangular matrix, the eigenvalues are the diagonal entries. Take A with rows (2, 0, 0), (1, 3, 0) and (4, −2, 5). The characteristic polynomial is λ³ − 10λ² + 31λ − 30, which factorises as (λ − 2)(λ − 3)(λ − 5), so the eigenvalues are 2, 3 and 5. The calculator gives the eigenvectors (1, −1, −2), (0, 1, 1) and (0, 0, 1).

Example 4: surd, repeated and complex eigenvalues

The calculator keeps surds exact and gives the decimals as well. Complex eigenvalues mean no direction stays on its own line, as with a rotation. A repeated eigenvalue can have one eigenvector direction or several: the 3 by 3 matrix with rows (4, 1, 0), (1, 4, 0) and (0, 0, 5) has λ = 5 repeated, with two independent eigenvectors, (1, 1, 0) and (0, 0, 1).

Check you have the 2 by 2 shortcut:

Useful facts for checking

• The eigenvalues add up to the trace, the main diagonal added together.
• The eigenvalues multiply to the determinant.
• A triangular or diagonal matrix has its diagonal entries as its eigenvalues.
• A symmetric matrix with real entries has real eigenvalues.
• Any non-zero multiple of an eigenvector is also an eigenvector, so (2, 1) and (4, 2) are the same answer.

Determinants are covered in our determinant calculator guide, and the rest of the Matrix Calculator in the matrix calculator guide.

Common mistakes

• Subtracting λ from every entry. λI only has λ on the diagonal, so subtract λ from the diagonal entries only.
• Expecting one fixed eigenvector. Eigenvectors are only defined up to a multiple. Yours may be a multiple of the calculator's.
• Using the zero vector. v = 0 always solves Av = λv, so it is never counted as an eigenvector.
• Sign errors in the determinant. Check that the eigenvalues add up to the trace.
• Trying a non-square matrix. Only square matrices have eigenvalues.

Practise linear algebra in a short timed paper

Eigenvalues and eigenvectors come up in our College Linear Algebra daily papers, which open as University Linear Algebra in the UK. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics.

Read next

Maths · Vectors and matricesVector calculator: magnitude, direction, add, dot and crossFree vector calculator: magnitude and direction, adding two vectors, the dot product, cross product and angle between them, in 2D or 3D.