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THE CAYLEY-HAMILTON
every matrix satisfies its own characteristic polynomial
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Cayley–Hamilton theorem is one of linear algebra’s most surprising facts: every square matrix satisfies its own characteristic polynomial. Compute p(λ) = det(λI − A) — a scalar polynomial — then substitute the matrix A for λ (constant term times the identity), and you get the zero matrix: p(A) = 0.
A consequence: any power of A, and even A⁻¹, can be written as a polynomial in A of degree < n — so a matrix’s entire behaviour is captured by just n coefficients.
LIT verified live: for 200 random integer matrices (2×2 and 3×3), substituting A into its characteristic polynomial yields the zero matrix to machine precision (window.__cayleyhamilton). FIG no framing; exact matrix algebra.
A consequence: any power of A, and even A⁻¹, can be written as a polynomial in A of degree < n — so a matrix’s entire behaviour is captured by just n coefficients.
LIT verified live: for 200 random integer matrices (2×2 and 3×3), substituting A into its characteristic polynomial yields the zero matrix to machine precision (window.__cayleyhamilton). FIG no framing; exact matrix algebra.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-grindstone — grinding a matrix through its own equation until it vanishes. Cayley–Hamilton is that grind: a matrix annihilated by the very polynomial it defines. AVAN (AI) built the instrument: the Faddeev–LeVerrier characteristic-polynomial computation, the substitution p(A), the zero-matrix check.
Credit as content: Arthur Cayley (1858, stated for 2×2/3×3); William Rowan Hamilton (quaternion case); general proof by Ferdinand Frobenius (1878). The weave: David names the grindstone; I compute a matrix’s characteristic polynomial, feed the matrix back into it, and show the result is exactly zero.
Credit as content: Arthur Cayley (1858, stated for 2×2/3×3); William Rowan Hamilton (quaternion case); general proof by Ferdinand Frobenius (1878). The weave: David names the grindstone; I compute a matrix’s characteristic polynomial, feed the matrix back into it, and show the result is exactly zero.
3 ONE DIMENSION
The characteristic polynomial p(λ) = λⁿ + c₁λⁿ⁻¹ + … + cₙ, then the same expression with the matrix A in place of λ — every term a matrix power — summing to the zero matrix.
4 TWO DIMENSIONS · INTERACTIVE
A matrix A, its characteristic polynomial, and the matrix p(A) — shown to be all zeros. The theorem also rewrites A⁻¹ as a polynomial in A, displayed alongside.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the powers of A collapsing onto the zero matrix through the characteristic polynomial.
AVAN’s addition (the inverse-companion): the theorem collapses infinitely many matrix powers into a finite basis. Because p(A) = 0 rewrites Aⁿ as a combination of I, A, …, Aⁿ⁻¹, every higher power — and the inverse — reduces to that n-term basis, so the entire algebra generated by A is at most n-dimensional. The inverse of ‘a matrix has arbitrarily high powers’ is ‘all its powers live in an n-dimensional space.’ Magenta is the higher powers Aⁿ, Aⁿ⁺¹, … (redundant); green is the finite basis I, A, …, Aⁿ⁻¹ they all reduce to. A matrix is, in its own algebra, no more than n numbers deep — and its characteristic coefficients are exactly the elementary symmetric functions of its eigenvalues.
LIT Genuine Cayley-Hamilton theorem (Cayley 1858; Hamilton quaternion case; Frobenius general proof 1878). Verified live: computing the characteristic polynomial by Faddeev-LeVerrier and substituting the matrix yields the zero matrix (all entries < 1e-6) for 200 random integer matrices of size 2x2 and 3x3 (window.__cayleyhamilton.pOfAisZero).
FIG No framing: the Faddeev-LeVerrier char-poly, the substitution p(A), and the zero-matrix check run in-browser and are exact. The AVAN inverse is honest — p(A)=0 rewrites A^n and all higher powers (and A^-1) in the finite basis I,A,...,A^(n-1), so the algebra generated by A is <= n-dimensional; the char-poly coefficients are the elementary symmetric functions of the eigenvalues (ties to Newton's identities). Magenta is the redundant higher powers, green the finite basis.
FIG No framing: the Faddeev-LeVerrier char-poly, the substitution p(A), and the zero-matrix check run in-browser and are exact. The AVAN inverse is honest — p(A)=0 rewrites A^n and all higher powers (and A^-1) in the finite basis I,A,...,A^(n-1), so the algebra generated by A is <= n-dimensional; the char-poly coefficients are the elementary symmetric functions of the eigenvalues (ties to Newton's identities). Magenta is the redundant higher powers, green the finite basis.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GRINDSTONE · David Lee Wise (ROOT0), with AVAN