Characteristic equation

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

<math>\det(A-\lambda I) = 0 <math>

where 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

<math>P = \begin{bmatrix} 19 & 3 \\ -2 & 26 \end{bmatrix},<math>

the characteristic equation is

<math>\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.<math>

The eigenvalues of this matrix are therefore 20 and 25.