Eigenvalue and Eigenvector Calculator
Find the eigenvalues, the eigenvectors and the characteristic polynomial of a 2 by 2 or 3 by 3 matrix.
What it does
Most matrices do one thing and one thing only: they stretch some directions and leave others alone. Those special directions are the eigenvectors, and the factors by which they are stretched are the eigenvalues. They are the reason a matrix can be understood, decomposed and turned into something easier to work with.
The University of Waterloo course notes behind this page give the recipe for a small matrix. The characteristic polynomial of a 2 by 2 matrix is lambda squared minus the trace times lambda plus the determinant, and the eigenvalues are the roots of it. For a 3 by 3 matrix the same notes expand the determinant into lambda cubed minus the trace times lambda squared, plus the sum of the principal minors times lambda, minus the determinant.
The Michigan lecture notes behind this page put the definition in the cleanest form. The characteristic polynomial of a matrix A is the determinant of A minus x times the identity, and a scalar lambda is an eigenvalue if and only if it is a root of that polynomial. The eigenvector for that root is any non zero vector that the matrix sends to lambda times itself.
This page does all of that for a 2 by 2 or a 3 by 3 matrix. Enter the matrix and it returns the characteristic polynomial, the eigenvalues, an eigenvector for each real one, the trace and the determinant, and it tells you whether the matrix can be diagonalised.
The limits are worth stating. The method is the closed form one for a quadratic and the cubic for the characteristic polynomial, and it is written for matrices of three rows or fewer. The eigenvector that comes out is one of infinitely many, since any multiple of it works just as well, and for a repeated eigenvalue the matrix may have fewer independent eigenvectors than the multiplicity suggests.
How to use
- Enter the matrix with one row per line, separated by semicolons.
- Use a 2 by 2 or a 3 by 3 matrix, which are the sizes the closed form handles.
- Read the characteristic polynomial, which is the equation the eigenvalues are the roots of.
- Read the eigenvalues, and check that they add up to the trace and multiply out to the determinant.
- Read the eigenvector for each real eigenvalue. Any multiple of it is just as good, so do not be surprised if the numbers look different from a textbook.
- Read the last row to see whether the matrix can be diagonalised, which needs as many independent eigenvectors as there are eigenvalues.
Formula
- A
- The matrix whose eigenvalues are being found.
- lambda
- An eigenvalue, which is a root of the characteristic polynomial.
- trace
- Sum of the entries on the main diagonal, which is also the sum of the eigenvalues.
- determinant
- Determinant of the matrix, which is also the product of the eigenvalues.
The eigenvector for a given eigenvalue is any non zero solution of the system A minus lambda times the identity times v equals zero, so it is only fixed up to a scale factor. When the eigenvalues are all real and all different the matrix can be diagonalised, and the Waterloo notes point out that a self adjoint matrix always has real eigenvalues.
Worked example
Example: the matrix 2 1; 1 2
Defaults: the two by two matrix with 2 and 1 on the diagonal and 1 in each off diagonal position.
- The trace is 2 + 2 = 4 and the determinant is 2 x 2 - 1 x 1 = 3.
- So the characteristic polynomial is lambda squared minus 4 lambda plus 3, which factors as (lambda - 3)(lambda - 1).
- The eigenvalues are therefore 3 and 1, and they add up to the trace and multiply to the determinant, as they must.
- For lambda = 3 the equations are -x + y = 0, so the eigenvector is (1, 1).
- For lambda = 1 the equations are x + y = 0, so the eigenvector is (-1, 1).
- Both eigenvalues are real and different, so the matrix can be diagonalised, and the two eigenvectors are independent.
- The tool shows the polynomial lambda squared - 4 lambda + 3, eigenvalues 3 and 1, eigenvectors (1, 1) and (-1, 1), a trace of 4 and a determinant of 3.
Reference table
| Matrix | Characteristic polynomial | Eigenvalues | Type |
|---|---|---|---|
| 2 1; 1 2 | lambda squared - 4 lambda + 3 | 3 and 1 | real and distinct |
| 2 0; 0 2 | lambda squared - 4 lambda + 4 | 2 and 2 | repeated |
| 1 2; 2 4 | lambda squared - 5 lambda + 0 | 5 and 0 | real and distinct |
| 0 1; 1 0 | lambda squared - 1 lambda + 0 | 1 and -1 | real and distinct |
Limits
- The method is the closed form one for a quadratic and a cubic, and it is written for matrices of three rows or fewer.
- The eigenvector that comes out is only fixed up to a scale factor, and any multiple of it works just as well.
- A repeated eigenvalue does not mean a repeated eigenvector, and a matrix with a repeated eigenvalue may not be diagonalisable at all.
- The numbers are exact only in the sense of the arithmetic, and a matrix with awkward entries gives awkward roots.
Common mistakes
- Expecting the eigenvalues to be the entries on the diagonal. That is only true for a triangular matrix.
- Using the row sums or the column sums instead of the determinant to build the characteristic polynomial.
- Forgetting the sign of the determinant term in the characteristic polynomial, which changes every root.
- Treating the eigenvector as a unique vector, when any multiple of it is an eigenvector too.
- Expecting a matrix with a repeated eigenvalue to have as many independent eigenvectors as the multiplicity, which is not always so.
Try it yourself
Quick checks. Answers sit tucked under each question.
What are the eigenvalues of the matrix 0 1; 1 0?
1 and -1. The trace is 0 and the determinant is -1, so the characteristic polynomial is lambda squared - 1 lambda + 0, whose roots are 1 and -1. The matrix swaps the two coordinates, so it leaves the line x equals y alone and reverses the line x equals minus y.
What happens when the eigenvalues are repeated?
The matrix 2 0; 0 2 has the characteristic polynomial lambda squared - 4 lambda + 4, with the single root 2 twice. The matrix is already diagonal, so it has two independent eigenvectors even though there is only one distinct eigenvalue.
How do I check the answer without solving anything?
Add the eigenvalues up and compare with the trace, then multiply them and compare with the determinant. For the example that is 3 + 1 = 4 and 3 x 1 = 3, which is exactly the trace and the determinant of the matrix.
Questions
What is an eigenvalue?
A number that a matrix multiplies one of its directions by. The Michigan notes define it as the scalar lambda for which there is a non zero vector v with A times v equal to lambda times v.
How is the characteristic polynomial built?
By taking the determinant of the matrix with lambda times the identity taken away from it. The Waterloo notes show that for a 2 by 2 matrix this expands to lambda squared minus the trace times lambda plus the determinant.
What does the trace tell me?
The sum of the eigenvalues, because the trace of a matrix is the sum of the entries on the diagonal and the characteristic polynomial picks that up. The same holds for the determinant and the product of the eigenvalues.
When can a matrix be diagonalised?
When it has as many independent eigenvectors as it has eigenvalues. The Waterloo notes show that a self adjoint matrix always has real eigenvalues, and the CMU notes make the point that a stationary distribution is a left eigenvector of eigenvalue one.
Why is the eigenvector not unique?
Because any multiple of it satisfies the same equation, and so does the negative of it. The tool normalises each one so that its largest component is one, which is a convention rather than a requirement.
Where do eigenvalues turn up outside of algebra?
Everywhere a linear process repeats. The Markov chain page on this site works with an eigenvalue of one, the oscillations of a coupled system are its eigenvectors, and the analysis of a structure in its own natural modes is the same calculation.
Sources
- University of Waterloo electrical engineering notes: determinants, characteristic polynomials and inverses
- University of Michigan Math 217 notes: computing eigenvalues and their multiplicities