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This online calculator computes the eigenvalues of a square matrix by solving the characteristic equation. The characteristic equation is the equation obtained by equating the characteristic polynomial to zero. Thus, this calculator first gets the characteristic equation using the 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 the The solution of a quadratic equation, Cubic equation and Quartic equation solution calculators. Thus it can find eigenvalues of a square matrix up to the fourth degree.
It is very unlikely that you would have a square matrix of a higher degree in math problems, because, according to the Abel–Ruffini theorem, a general polynomial equation of degree five or higher has no solution in radicals, thus, it can be solved only by numerical methods. (Note that the degree of a characteristic polynomial is the degree of its square matrix). More theory can be found below the calculator.
Eigenvalues are easier to explain with eigenvectors. Suppose we have a square matrix A. This matrix defines a linear transformation, that is, if we multiply any vector by A, we get the new vector that 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 characteristic equation of A, and the left part of it is called the characteristic polynomial of A.
The roots of this equation are eigenvalues of A, also called characteristic values, or characteristic roots.
The characteristic equation of A is a polynomial equation, and to get polynomial coefficients you need to expand the determinant of matrix
For a 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 the Faddeev–LeVerrier algorithm as it is done in the Characteristic polynomial calculator.
Once you get the characteristic equation in polynomial form, you can solve it for eigenvalues. And here you can find an excellent introduction as to why we would ever care about finding eigenvalues and eigenvectors, and why they are very important concepts in linear algebra.