Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
For , the characteristic polynomial is when , with for the unique matrix
Definition
Let be a field and let . If , regard as a matrix over the commutative polynomial ring . The characteristic polynomial of is
For , define the characteristic polynomial of the unique matrix to be the constant polynomial . This agrees with the empty-product convention and makes the characteristic polynomial monic of degree in the zero-sized case.
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
- 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
- Computing the minimal polynomial of an explicit 3×3 idempotent from its annihilating polynomials Example
- The matrix beginpmatrix2&10&3 endpmatrix has characteristic polynomial (x-2)(x-3) and two explicitly computed eigenspaces Example
- The characteristic polynomial of a block upper- or lower-triangular matrix is the product of the characteristic polynomials of its diagonal blocks 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
- For A,B∈ Mₙ(F), the products AB and BA have the same characteristic polynomial Theorem
- Similar matrices have the same characteristic polynomial 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)