Matrix Algebra

Characteristic Equation

Characteristic equation. Observe that an eigenvector is a nonzero vector $\mathbf{x}$ satisfying $A\mathbf{x}=\lambda\mathbf{x}$ . Thus, a scalar value $\lambda$ is an eigenvector if the corresponding homogeneous equation $(A - \lambda I)\mathbf{x} = \mathbf{0}$ has nontrivial solutions. It is equivalently characterized by the fact that the matrix $(A - \lambda I)$ is not invertible. Thus, the scalar $\lambda$ is an eigenvalue of

if and only if it satisfies the characteristic equation

$\displaystyle \det(A - \lambda I) = 0.$

The characteristic equation is a polynomial equation of degree

, and the roots (the solutions) to the equation correspond to the eigenvalues of

. The multiplicity of a root is called the algebraic multiplicity of the corresponding eigenvalue $\lambda$ . Note that the geometric multiplicity can never exceed the algebraic multiplicity.

EXAMPLE 3. Construct the characteristic equation, and then find the eigenvalues for each of the following matrices.

$A = \begin{bmatrix} 2 & 3 \\ 3 &-6 \end{bmatrix}$
$A = \begin{bmatrix} 5 &-2 & 6 &-1 \\ 0 & 3 &-8 & 0 \\ 0 & 0 & 5 & 4 \\ 0 & 0 & 0 & 1 \end{bmatrix}$