Characteristic equation
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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.