The Characteristic Polynomial and Eigenvalues
Eigenvalues are precisely the roots of the polynomial equation obtained by setting the determinant of A minus lambda times the identity to zero.
From Vectors to Determinants
The defining equation for an eigenvector states that the matrix A acting on vector v equals lambda times v. Rearranging terms yields the homogeneous linear system (A - lambda I)v = 0. For a non-zero vector v to satisfy this system, the matrix (A - lambda I) must be non-invertible, meaning its null space is non-trivial.
(A - \lambda I)\mathbf{v} = \mathbf{0}The Characteristic Equation
A square matrix is singular exactly when its determinant vanishes. Taking the determinant of (A - lambda I) converts a geometric problem about vector directions into a scalar algebraic equation. Expanding this determinant produces an n-th degree polynomial in lambda whose roots are the eigenvalues of A.
p(\lambda) = \det(A - \lambda I) = 0Multiplicity and Spectrum
By the Fundamental Theorem of Algebra, an n-by-n complex matrix has exactly n eigenvalues counted with algebraic multiplicity. The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial. Even when real matrices yield complex conjugate roots, the polynomial fully captures the matrix spectrum.
p(\lambda) = (-1)^n (\lambda - \lambda_1)^{m_1} \cdots (\lambda - \lambda_k)^{m_k}Flashcard
The characteristic equation used to solve for matrix eigenvalues
Check your understanding
Why must the determinant of (A - lambda I) equal zero for lambda to be an eigenvalue of A?