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Characteristic Equation



From Eigenvectors to the Determinant Condition
The Characteristic Polynomial
Computing the Characteristic Polynomial: 2×2
Computing the Characteristic Polynomial: 3×3
Larger Matrices
Algebraic Multiplicity
Finding Eigenvectors After Finding Eigenvalues
The Cayley-Hamilton Theorem
Characteristic Polynomial and Similarity
Summary: The Characteristic Polynomial at a Glance



The Polynomial Whose Roots Are the Eigenvalues

The eigenvalue problem Av = λv converts into a determinant condition: det(A − λI) = 0. This determinant is a polynomial in λ whose roots are the eigenvalues. Computing the characteristic polynomial and factoring it is the standard method for finding eigenvalues of small matrices — and the polynomial's coefficients encode the trace, determinant, and other invariants of the matrix.



From Eigenvectors to the Determinant Condition

The equation Av=λvA\mathbf{v} = \lambda\mathbf{v} rearranges to (A−λI)v=0(A - \lambda I)\mathbf{v} = \mathbf{0}. This is a homogeneous system, and eigenvectors are its nontrivial solutions. Nontrivial solutions exist if and only if the coefficient matrix A−λIA - \lambda I is singular:

det⁡(A−λI)=0\det(A - \lambda I) = 0


This is the characteristic equation. It holds for exactly those values of λ\lambda that are eigenvalues of AA. Every other value of λ\lambda makes A−λIA - \lambda I invertible, the system has only the trivial solution, and no eigenvector exists for that λ\lambda.

The characteristic equation transforms the geometric question "which directions does AA preserve?" into the algebraic question "for which λ\lambda is this determinant zero?"

The Characteristic Polynomial

The expression p(λ)=det⁡(A−λI)p(\lambda) = \det(A - \lambda I) is a polynomial of degree nn in the variable λ\lambda. It is called the characteristic polynomial of AA.

For an n×nn \times n matrix, p(λ)p(\lambda) has degree nn with leading term (−1)nλn(-1)^n \lambda^n. The constant term is p(0)=det⁡(A)p(0) = \det(A) — the determinant of the matrix itself. The coefficient of λn−1\lambda^{n-1} is (−1)n−1tr(A)(-1)^{n-1}\text{tr}(A), connecting the next-to-leading term to the trace.

The eigenvalues are precisely the roots of p(λ)=0p(\lambda) = 0. Every root is an eigenvalue, and every eigenvalue is a root. The characteristic polynomial packages the entire eigenvalue structure of the matrix into a single algebraic expression.

Computing the Characteristic Polynomial: 2×2

For A=(abcd)A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, the characteristic polynomial is

p(λ)=det⁡(a−λbcd−λ)=(a−λ)(d−λ)−bc=λ2−(a+d)λ+(ad−bc)p(\lambda) = \det\begin{pmatrix} a - \lambda & b \\ c & d - \lambda \end{pmatrix} = (a - \lambda)(d - \lambda) - bc = \lambda^2 - (a + d)\lambda + (ad - bc)


This is λ2−tr(A)λ+det⁡(A)\lambda^2 - \text{tr}(A)\lambda + \det(A). The eigenvalues follow from the quadratic formula:

λ=tr(A)±tr(A)2−4det⁡(A)2\lambda = \frac{\text{tr}(A) \pm \sqrt{\text{tr}(A)^2 - 4\det(A)}}{2}


The discriminant Δ=tr(A)2−4det⁡(A)\Delta = \text{tr}(A)^2 - 4\det(A) classifies the eigenvalue type. When Δ>0\Delta > 0, there are two distinct real eigenvalues. When Δ=0\Delta = 0, there is one repeated real eigenvalue. When Δ<0\Delta < 0, the eigenvalues are a complex conjugate pair.

Worked Example


For A=(5234)A = \begin{pmatrix} 5 & 2 \\ 3 & 4 \end{pmatrix}: tr(A)=9\text{tr}(A) = 9, det⁡(A)=14\det(A) = 14, Δ=81−56=25\Delta = 81 - 56 = 25. The eigenvalues are λ=9±52\lambda = \frac{9 \pm 5}{2}, giving λ1=7\lambda_1 = 7 and λ2=2\lambda_2 = 2.
Discriminant Δ = tr(A)² − 4·det(A) Eigenvalue type Independent real eigenvectors
Δ > 0 two distinct real eigenvalues 2 — always diagonalizable
Δ = 0 one repeated real eigenvalue 1 or 2 — diagonalizable only when A = λ · I (otherwise defective)
Δ < 0 complex conjugate pair a ± bi 0 real (one conjugate pair of complex eigenvectors over ℂ)

Computing the Characteristic Polynomial: 3×3

For a 3×33 \times 3 matrix, expanding det⁡(A−λI)\det(A - \lambda I) using cofactor expansion produces a cubic polynomial:

p(λ)=−λ3+tr(A)λ2−(sum of 2×2 principal minors)λ+det⁡(A)p(\lambda) = -\lambda^3 + \text{tr}(A)\lambda^2 - (\text{sum of } 2 \times 2 \text{ principal minors})\lambda + \det(A)


The computation is lengthier but follows the same cofactor mechanics as any 3×33 \times 3 determinant.

Worked Example


For A=(210031001)A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & 1 \\ 0 & 0 & 1 \end{pmatrix}, this is upper triangular, so A−λIA - \lambda I is also upper triangular with diagonal entries 2−λ2 - \lambda, 3−λ3 - \lambda, 1−λ1 - \lambda:

p(λ)=(2−λ)(3−λ)(1−λ)p(\lambda) = (2 - \lambda)(3 - \lambda)(1 - \lambda)


The eigenvalues are λ=1,2,3\lambda = 1, 2, 3 — readable directly from the diagonal. For triangular matrices, the characteristic polynomial always factors as the product of the diagonal terms, making the eigenvalues visible by inspection.

For non-triangular 3×33 \times 3 matrices, the cubic must be factored by finding rational roots (testing factors of the constant term), by inspection, or by the cubic formula.

Larger Matrices

For an n×nn \times n matrix, the characteristic polynomial has degree nn, and finding its roots becomes increasingly difficult as nn grows. There is no general closed-form formula for roots of polynomials of degree 55 or higher (Abel-Ruffini theorem), so explicit factoring is limited to small matrices or matrices with special structure.

Diagonal and triangular matrices are immediate: the eigenvalues are the diagonal entries. Block triangular matrices factor block by block: the characteristic polynomial is the product of the characteristic polynomials of the diagonal blocks.

For general large matrices, eigenvalues are computed numerically by iterative algorithms — most importantly the QR algorithm, which repeatedly applies QR decompositions to converge on the eigenvalues without ever forming the characteristic polynomial explicitly. Computing the polynomial and then finding its roots is numerically unstable for large nn and is never used in practice.
Matrix structure How to find the eigenvalues
Diagonal eigenvalues are the diagonal entries — read off by inspection
Triangular (upper or lower) eigenvalues are the diagonal entries (det of a triangular matrix is the product of its diagonal)
Block triangular multiply the characteristic polynomials of each diagonal block; eigenvalues are the union of block eigenvalues
General 2×2 quadratic formula on λ² − tr(A)·λ + det(A) = 0
General 3×3 or 4×4 expand det(A − λI) and factor (rational-root testing, cubic/quartic formulas, or inspection)
General n ≥ 5 (numerical) iterative algorithms such as QR iteration — the polynomial is never formed explicitly

Algebraic Multiplicity

If λ0\lambda_0 is a root of the characteristic polynomial p(λ)p(\lambda), its algebraic multiplicity is the largest power kk such that (λ−λ0)k(\lambda - \lambda_0)^k divides p(λ)p(\lambda). Equivalently, it is the multiplicity of λ0\lambda_0 as a root.

If p(λ)=(λ−2)3(λ+1)p(\lambda) = (\lambda - 2)^3(\lambda + 1), then λ=2\lambda = 2 has algebraic multiplicity 33 and λ=−1\lambda = -1 has algebraic multiplicity 11. The algebraic multiplicities of all eigenvalues sum to nn — the degree of the polynomial — when complex roots are included.

The algebraic multiplicity is an upper bound for the geometric multiplicity: 1≤mg(λ)≤ma(λ)1 \leq m_g(\lambda) \leq m_a(\lambda). The geometric multiplicity is the dimension of the eigenspace, and it can be strictly smaller than the algebraic multiplicity. When this gap occurs for any eigenvalue, the matrix is not diagonalizable.

Finding Eigenvectors After Finding Eigenvalues

Once the eigenvalues are known, the eigenvectors for each λi\lambda_i are found by solving the homogeneous system (A−λiI)v=0(A - \lambda_i I)\mathbf{v} = \mathbf{0}.

Row reduce A−λiIA - \lambda_i I and express the general solution in parametric form. Each free variable contributes one basis vector for the eigenspace EλiE_{\lambda_i}.

Worked Example


For A=(1243)A = \begin{pmatrix} 1 & 2 \\ 4 & 3 \end{pmatrix}, the characteristic polynomial is λ2−4λ−5=(λ−5)(λ+1)\lambda^2 - 4\lambda - 5 = (\lambda - 5)(\lambda + 1). Eigenvalues: λ1=5\lambda_1 = 5, λ2=−1\lambda_2 = -1.

For λ1=5\lambda_1 = 5: A−5I=(−424−2)A - 5I = \begin{pmatrix} -4 & 2 \\ 4 & -2 \end{pmatrix}. Row reducing: (1−1/200)\begin{pmatrix} 1 & -1/2 \\ 0 & 0 \end{pmatrix}. Free variable v2=tv_2 = t, so v1=t/2v_1 = t/2. Eigenvector: v1=(1,2)T\mathbf{v}_1 = (1, 2)^T.

For λ2=−1\lambda_2 = -1: A+I=(2244)A + I = \begin{pmatrix} 2 & 2 \\ 4 & 4 \end{pmatrix}. Row reducing: (1100)\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}. Free variable v2=tv_2 = t, so v1=−tv_1 = -t. Eigenvector: v2=(−1,1)T\mathbf{v}_2 = (-1, 1)^T.

Verification: Av1=(510)=5v1A\mathbf{v}_1 = \begin{pmatrix} 5 \\ 10 \end{pmatrix} = 5\mathbf{v}_1 and Av2=(1−1)=−1⋅v2A\mathbf{v}_2 = \begin{pmatrix} 1 \\ -1 \end{pmatrix} = -1 \cdot \mathbf{v}_2.

The Cayley-Hamilton Theorem

Every square matrix satisfies its own characteristic polynomial. If p(λ)=det⁡(A−λI)p(\lambda) = \det(A - \lambda I) is the characteristic polynomial, then

p(A)=0p(A) = 0


where 00 is the zero matrix and λ\lambda is replaced by AA (with constant terms multiplied by II).

For example, if p(λ)=λ2−5λ+6p(\lambda) = \lambda^2 - 5\lambda + 6, then A2−5A+6I=OA^2 - 5A + 6I = O. This can be rearranged to express A−1A^{-1} as a polynomial in AA: A−1=16(5I−A)A^{-1} = \frac{1}{6}(5I - A) (provided det⁡(A)=6≠0\det(A) = 6 \neq 0). More generally, the Cayley-Hamilton theorem guarantees that A−1A^{-1} can always be written as a polynomial in AA of degree at most n−1n - 1.

The theorem also shows that any power AkA^k with k≥nk \geq n can be reduced to a polynomial in AA of degree at most n−1n - 1 — the characteristic polynomial provides a recurrence that expresses higher powers in terms of lower ones.

Characteristic Polynomial and Similarity

Similar matrices have the same characteristic polynomial:

det⁡(P−1AP−λI)=det⁡(P−1(A−λI)P)=det⁡(A−λI)\det(P^{-1}AP - \lambda I) = \det(P^{-1}(A - \lambda I)P) = \det(A - \lambda I)


The second equality uses the multiplicative property of the determinant: det⁡(P−1)det⁡(A−λI)det⁡(P)=det⁡(A−λI)\det(P^{-1})\det(A - \lambda I)\det(P) = \det(A - \lambda I), since det⁡(P−1)det⁡(P)=1\det(P^{-1})\det(P) = 1.

This means the characteristic polynomial is a property of the linear transformation itself, not of any particular matrix representation. Changing the basis changes the matrix but not the polynomial. Since the eigenvalues are the roots of the polynomial, similar matrices have the same eigenvalues with the same algebraic multiplicities.

The trace and determinant are just two of the nn coefficients of the characteristic polynomial. The polynomial carries more information than either one alone — it determines the complete multiset of eigenvalues, not just their sum and product.

Summary: The Characteristic Polynomial at a Glance

The characteristic polynomial p(λ) = det(A − λI) carries every spectral fact about A — its degree, individual coefficients, roots, and behavior under similarity all encode information about the matrix. The table below collects each structural property of the polynomial alongside its statement and the connection it makes to A.
Aspect of p(λ) = det(A − λI), for n × n A Statement Connection
Degree deg p = n matches the matrix size
Leading term (−1)n λn sign alternates with n; sometimes flipped to make it monic
Coefficient of λn−1 (−1)n−1 · tr(A) sum of eigenvalues equals tr(A)
Constant term p(0) det(A) product of eigenvalues equals det(A); A invertible ⟺ p(0) ≠ 0
Roots the n eigenvalues with algebraic multiplicity spectral characterization of A
Similarity similar matrices share p(λ) p is a basis-independent invariant of the transformation
Cayley-Hamilton p(A) = O (zero matrix) A−1 and high powers Ak become polynomials in A of degree ≤ n − 1
2×2 form p(λ) = λ² − tr(A) · λ + det(A) discriminant Δ = tr² − 4·det classifies eigenvalue type
3×3 form p(λ) = −λ³ + tr(A) · λ² − M₂ · λ + det(A) M₂ is the sum of 2×2 principal minors