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Characteristic equation

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

det(A - λ I) = 0

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

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

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.

Categories: Linear algebra | Equations

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10-26-2009 08:16:03
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Characteristic equation