This online calculator computes the eigenvalues of a square matrix by solving the characteristic equation. The characteristic equation is the equation obtained by equating to zero the characteristic polynomial. Thus, this calculator first gets the characteristic equation using Characteristic polynomial calculator, then solves it analytically to obtain eigenvalues (either real or complex). It does so only for matrices 2x2, 3x3, and 4x4, using Solution of quadratic equation, Cubic equation and Quartic equation solution calculators. Thus it can find eigenvalues of a square matrix up to 4th degree.
It is very unlikely that you have square matrix of higher degree in math problems, because, according to Abel–Ruffini theorem, a general polynomial equation of degree 5 or higher has no solution in radicals, thus, it can be solved only by numerical methods. (Note that degree of characteristic polynomial is the degree of its square matrix). More theory can be found below the calculator.
Eigenvalues are better to explain with the eigenvectors. Suppose we have a square matrix A. This matrix defines linear transformation, that it, if we multiply any vector by A, we get the new vector which changes direction:
However, there are some vectors for which this transformation produces the vector that is parallel to the original vector. In other words:
where is some scalar number.
These vectors are eigenvectors of A, and these numbers are eigenvalues of A.
This equation can be rewritten as
where I is the identity matrix.
Since v is non-zero, the matrix is singular, which means that its determinant is zero.
is the characteric equation of A, and the left part of it is called characteric polynomial of A.
The roots of this equation are eigenvalues of A, also called characteristic values, or characteristic roots.
Characteric equation of A is a polynomial equation, and to get polynomial coefficients you need to expand the determinant of matrix
For 2x2 case we have a simple formula:
where trA is the trace of A (sum of its diagonal elements) and detA is the determinant of A. That is
For other cases you can use Faddeev–LeVerrier algorithm as it is done in Characteristic polynomial calculator.
Once you get the characteric equation in polynomial form, you can solve it for eigenvalues. And here you can find excellent introduction of why we ever care for finding eigenvalues and eigenvectors, and why they are very important concepts in linear algebra.