For $A\in M_n(F)$, the characteristic polynomial is ${\chi }_A(x)=\det (x{I}_n-A)$ when $n\ge 1$, with ${\chi }_A(x)=1$ for the unique $0\times 0$ matrix

Depends on

• For $n\ge1$, the determinant over a commutative ring by the Leibniz formula, and $|\det A|$ for a real matrix
• The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution
• Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose
• For a field, the ring-matrix operations, invertibility and similarity agree exactly with the established field-matrix interface

Used by

• For an invertible matrix over a field, Cayley-Hamilton makes every matrix-power entry and trace sequence linearly recurrent Corollary
• Coercive non-symmetric forms need not have an orthonormal eigenbasis Counterexample
• Adjacency spectrum, spectral radius, and cospectral graphs Definition
• The basis-independent characteristic polynomial χ_T of an endomorphism of a finite-dimensional space, including χ_T=1 in dimension zero Definition
• A coercive non-symmetric form can have non-real Galerkin eigenvalues Example
• Computing the minimal polynomial of an explicit 3×3 idempotent from its annihilating polynomials Example
• The exceptional spectra for E₆ and H₃ computed exactly: characteristic polynomials, cyclotomic factorisations and the resulting degree tables Example
• The matrix beginpmatrix2&10&3 endpmatrix has characteristic polynomial (x-2)(x-3) and two explicitly computed eigenspaces Example
• A Hermitian two-by-two pivot inertia is determined by determinant and trace signs Lemma
• A simple neutral multiplier and transverse phase make the shooting Jacobian invertible Lemma
• Characteristic roots preserve separated cluster counts under matrix perturbation Lemma
• Constructible images for finite-presentation affine maps Lemma
• Fixed-memory L-Bfgs curvature and scale bounds imply uniform spectral bounds Lemma
• Powers of a Schur-stable matrix decay geometrically up to a polynomial factor Lemma
• The characteristic polynomial of a block upper- or lower-triangular matrix is the product of the characteristic polynomials of its diagonal blocks Lemma
• The Coxeter elements of the classical types Aₙ, Bₙ, Dₙ and I₂(m): characteristic polynomials, orders and spectral exponents from their reflection models Lemma
• The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness Lemma
• The six exceptional Coxeter spectra: characteristic polynomials, orders and spectral exponents from exact matrices Lemma
• χ_A(x) is monic of degree n; for n≥1 its xⁿ⁻¹ coefficient is -tr(A) and its constant coefficient is (-1)ⁿ det(A), while χ_0×0=1 Lemma
• The adjacency spectrum is an isomorphism invariant Proposition
• A recurrence companion matrix has the recurrence characteristic polynomial Theorem
• A regular Coxeter eigenvector determines the basic degrees: the exponent-residue identification and the complete degree tables for all finite Coxeter types Theorem
• A Sylvester equation is uniquely solvable exactly when the spectra are disjoint Theorem
• For A,B∈ Mₙ(F), the products AB and BA have the same characteristic polynomial Theorem
• Similar matrices have the same characteristic polynomial Theorem
• The total degree sum, the invariant Jacobian as the discriminant, anti-invariants, and the top coinvariant class Theorem

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• H. Pinkham, Linear Algebra, §§12.1–12.4 (standard reference, not scraped)
• M. Khovanov, Linear Algebra II notes, §6 (standard reference, not scraped)