Characteristic equation

From Wikipedia, the free encyclopedia

In linear algebra, the characteristic equation (or secular equation) of a square matrix A is the equation in one variable λ

$\det(A-\lambda I) = 0 \,$

where $det$ is the determinant and $I$ is the identity matrix. The solutions of the characteristic equation are precisely the eigenvalues of the matrix A. The polynomial to the left of "=" is the characteristic polynomial of the matrix.

For example, for the matrix

$P = \begin{bmatrix} 19 & 3 \\ -2 & 26 \end{bmatrix},$

the characteristic equation is

$\det(P - \lambda I) = \det\begin{bmatrix} 19-\lambda & 3 \\ -2 & 26-\lambda \end{bmatrix} =\lambda^2-45\lambda+500=(\lambda-25)(\lambda-20)=0.$

The eigenvalues of this matrix are therefore 20 and 25.

Some shortcuts exist for low dimension matrices. For a 2x2 matrix A, the characteristic equation can be found from its determinant and trace (Tr(A)) to be