Characteristic Polynomial

Definition

Matrix Polynomial

Characteristic polynomial: a polynomial associated with a square matrix A. Form: p(λ) = det(λI - A). Degree: n for n×n matrix.

Variable and Coefficients

Variable λ: scalar parameter. Coefficients: functions of matrix entries, typically real or complex.

Domain and Codomain

Polynomial over field F (commonly ℝ or ℂ). Maps scalar λ to determinant in F.

Construction

Identity Matrix Scale

Form λI: scalar λ multiplied by identity matrix. Ensures dimension and operator compatibility.

Matrix Subtraction

Subtraction λI - A: matrix polynomial in λ. Results in matrix-valued polynomial.

Determinant Operation

Determinant det(λI - A): scalar polynomial. Encodes eigenvalue information.

p(λ) = det(λI - A)

Properties

Degree and Leading Coefficient

Degree = n (size of matrix). Leading coefficient = 1 (monic polynomial).

Coefficient Relationships

Constant term = (−1)ⁿ det(A). Coefficient of λ^{n-1} = −tr(A).

Invariant under Similarity

Characteristic polynomial unchanged by similarity transformations: p_A(λ) = p_{P^{-1}AP}(λ).

Real and Complex Roots

Roots correspond to eigenvalues; may be complex even if matrix is real.

Relation to Eigenvalues

Definition of Eigenvalues

Eigenvalues λ satisfy (A - λI)v = 0 for nonzero vector v.

Roots of Characteristic Polynomial

Eigenvalues = roots of p(λ) = 0. Multiplicity corresponds to algebraic multiplicity.

Geometric vs Algebraic Multiplicity

Algebraic multiplicity: multiplicity as root of p(λ). Geometric multiplicity: dimension of eigenspace.

Spectrum of Matrix

Spectrum = set of eigenvalues = roots of characteristic polynomial.

Computational Methods

Determinant Expansion

Use Laplace expansion or cofactor expansion for det(λI - A). Computationally expensive for large n.

Leverrier-Faddeev Algorithm

Recursive formula to compute coefficients using traces of powers of A.

Use of Eigenvalue Algorithms

Numerical methods (QR algorithm) compute eigenvalues, indirectly giving roots of characteristic polynomial.

Symbolic Computation

Computer algebra systems (Mathematica, Maple) calculate characteristic polynomial symbolically.

Let A be n×n matrix:Initialize B_0 = IFor k=1 to n: c_k = -(1/k) * tr(A*B_{k-1}) B_k = A*B_{k-1} + c_k*ICharacteristic polynomial:p(λ) = λ^n + c_1 λ^{n-1} + ... + c_n

Examples

2×2 Matrix

Matrix: A = [[a, b], [c, d]]. Characteristic polynomial: p(λ) = λ² - (a+d)λ + (ad - bc).

3×3 Matrix

Matrix: A = [[a, b, c], [d, e, f], [g, h, i]]. Polynomial degree 3 with coefficients from traces and determinants.

Diagonal Matrix

Eigenvalues = diagonal entries. Characteristic polynomial: product of (λ - a_ii).

Nilpotent Matrix

All eigenvalues zero. Characteristic polynomial: λ^n.

Algebraic Multiplicity

Definition

Multiplicity of eigenvalue as root of characteristic polynomial.

Examples

Eigenvalue λ=3 with multiplicity 2 implies (λ - 3)² divides p(λ).

Relation to Jordan Form

Algebraic multiplicity ≥ geometric multiplicity. Determines size of Jordan blocks.

Implications

Multiplicity affects diagonalizability and eigenstructure complexity.

Cayley-Hamilton Theorem

Statement

Every square matrix satisfies its own characteristic polynomial.

Formal Expression

p(A) = 0

Consequences

Used to compute matrix functions, powers, and minimal polynomial relations.

Proof Sketch

Via adjugate matrix properties or polynomial factorization over algebraic closure.

Applications

Eigenvalue Determination

Roots of characteristic polynomial identify eigenvalues critical in systems analysis.

Diagonalization Criteria

Multiplicity and roots guide diagonalizability and canonical form derivation.

Control Theory

Characteristic polynomial used in stability analysis of dynamical systems.

Quantum Mechanics

Operator spectra linked to characteristic polynomial roots.

Characteristic Polynomial of Linear Transformations

Definition

For linear operator T: V → V, polynomial p(λ) = det(λI - [T]_B).

Basis Independence

Polynomial invariant under choice of basis due to similarity invariance.

Representation

Matrix representation required to compute polynomial explicitly.

Extension to Infinite Dimensions

Characteristic polynomial defined only for finite-dimensional linear operators.

Generalizations

Minimal Polynomial

Minimal polynomial divides characteristic polynomial; captures minimal annihilating polynomial.

Characteristic Polynomial over Rings

Defined for matrices over commutative rings with unity; properties vary.

Multilinear Algebra Extensions

Characteristic polynomial concept extends to tensor algebra and multilinear maps.

Pencils and Parameter-Dependent Polynomials

Generalized characteristic polynomials appear in matrix pencils and parameterized systems.

Limitations and Caveats

Computational Complexity

Determinant calculation expensive for large matrices; numerical methods preferred.

Numerical Stability

Characteristic polynomial sensitive to perturbations; eigenvalue computations may be unstable.

Non-Uniqueness of Eigenvectors

Polynomial roots do not determine eigenvectors uniquely; geometric multiplicity varies.

Not Always Diagonalizable

Repeated roots may correspond to defective matrices; Jordan form needed.

References

• Horn, R. A., & Johnson, C. R. "Matrix Analysis." Cambridge University Press, vol. 2, 2013, pp. 45-95.
• Strang, G. "Introduction to Linear Algebra." Wellesley-Cambridge Press, 5th ed., 2016, pp. 200-245.
• Axler, S. "Linear Algebra Done Right." Springer, 3rd ed., 2015, pp. 120-160.
• Lay, D. C. "Linear Algebra and Its Applications." Pearson, 5th ed., 2015, pp. 350-400.
• Shilov, G. E. "Linear Algebra." Dover Publications, 1977, pp. 78-110.