The Characteristic Equation

For a matrix that is not triangular, the eigenvalues do not sit on the diagonal and must be found some other way. A scalar $λ$ is an eigenvalue of $A$ exactly when $A - λ I$ is singular, and singularity is detected by the determinant. This turns the search for eigenvalues into a single polynomial equation in $λ$ .

The determinant condition

A matrix fails to be invertible exactly when its determinant is zero, so the eigenvalues of $A$ are the scalars $λ$ satisfying $det (A - λ I) = 0$ .

The determinant has converted the matrix equation $(A - λ I) x = 0$ , which carries two unknowns $λ$ and $x$ , into a scalar equation in the single unknown $λ$ .

The characteristic polynomial

det (A - λ I)

, plotted here as

p (t)

, crosses zero at each eigenvalue; the roots sit at

- 7

and

3

Determinant by row reduction

For matrices larger than $2 \times 2$ , the determinant is computed by row reduction. Reduce $A$ to an echelon form $U$ using only row replacements and row interchanges (no scaling), and let $r$ be the number of interchanges. Then

det A = {(- 1)^{r} \cdot (product of the pivots in U), 0, A invertible, A not invertible .

The first property adds two more equivalent conditions to the Invertible Matrix Theorem.

Multiplicity

The characteristic polynomial of an $n \times n$ matrix has degree $n$ , so counting multiplicities and allowing complex roots it has exactly $n$ roots. A root can repeat.

For the upper-triangular matrix

A = 5000 - 2 300 6 - 8 50 - 1 041,

Part (d) of the determinant-properties theorem gives $det (A - λ I) = (5 - λ)^{2} (3 - λ) (1 - λ)$ . The eigenvalue $5$ has multiplicity $2$ ; the eigenvalues $3$ and $1$ each have multiplicity $1$ . As another example, a $6 \times 6$ matrix with characteristic polynomial

λ^{6} - 4 λ^{5} - 12 λ^{4} = λ^{4} (λ - 6) (λ + 2)

has eigenvalues $0$ (multiplicity $4$ ), $6$ , and $- 2$ , which can be listed with repetition as $0, 0, 0, 0, 6, - 2$ .

A simple root crosses the horizontal axis; a repeated root touches it without crossing, its multiplicity visible as tangency.

Similarity

The characteristic polynomial is a property of the transformation, not of the particular coordinates chosen to represent it. Similarity makes this precise.

A similarity transformation reads

A

in the coordinate frame set by

P

; the change of frame does not disturb the eigenvalues.

Two warnings:

Equal eigenvalues do not imply similarity. The matrices $[2012]$ and $[2002]$ share the eigenvalue $2$ (twice) but are not similar; the second is $2 I$ , similar only to itself.
Similarity is not row equivalence. Row operations correspond to left-multiplication $B = E A$ , not to $P^{- 1} A P$ , and they usually change the eigenvalues.

The similarity-invariance theorem is the foundation of the practical eigenvalue algorithms: they generate a sequence of matrices all similar to $A$ (hence sharing its eigenvalues) that drift toward triangular form, whose diagonal then displays the eigenvalues.

A dynamical system in closed form

Similarity and the characteristic equation combine to solve a difference equation in closed form.

Object	Role
$det (A - λ I)$	characteristic polynomial, degree $n$
roots of $det (A - λ I) = 0$	the eigenvalues
multiplicity of a root	algebraic multiplicity of that eigenvalue
$λ = 1$ eigenvector	steady-state direction
$∣ λ ∣ < 1$ eigenvalues	transient parts, decaying as $λ^{k}$

The decomposition $x_{0} = c_{1} v_{1} + c_{2} v_{2}$ works only when $A$ has a full set of independent eigenvectors. That condition, restated as a matrix factorization, is diagonalization.¹

Lay, Linear Algebra and Its Applications, 5th ed., §5.2 — The Characteristic Equation: $det (A - λ I) = 0$ , the characteristic polynomial and algebraic multiplicity, the determinant/eigenvalue additions to the Invertible Matrix Theorem, and Theorem 4 on similar matrices sharing a characteristic polynomial. ↩