Crouzeix's Conjecture

Feed a matrix into a polynomial. How much bigger can the answer get than the polynomial itself?

Posed by Michel Crouzeix, 2004.

In plain terms

Take a square matrix A and a polynomial p, and form p(A) by substituting the matrix for the variable. How large can p(A) be? There is an obvious region of the complex plane attached to A — its numerical range, the set of all quantities x*Ax over unit vectors x, which is a convex blob containing the eigenvalues. If p is small everywhere on that blob, must p(A) be small too?

Not exactly: there is a factor to pay. Crouzeix conjectured the factor is 2 — that p(A) can never exceed twice the largest size of p on the blob. Two is certainly the right guess for the smallest possible factor, because a single 2×2 matrix already achieves it exactly: the matrix with a lone 1 above the diagonal has size 1, while its numerical range is a disc of radius one half.

The conjecture says that little example is as bad as it ever gets, in any dimension, for any polynomial.

What is known

The factor is somewhere between 2 and 1 + √2 ≈ 2.414. Crouzeix proved 11.08 in 2007; Crouzeix and Palencia brought it down to 1 + √2 in 2017, and that has stood since. Nobody has closed the remaining gap of about four tenths.

Special cases fall regularly. It is settled for 2×2 matrices, for normal matrices, for matrices whose numerical range happens to be a disc, for matrices close to a single Jordan block, for certain 3×3 tridiagonal matrices, and for weighted shifts. Recent work has produced constants better than 1 + √2 for each fixed dimension separately, and has reformulated the general conjecture as a statement about a single differentiation operator — so the whole problem is equivalent to one concrete question about analytic functions.

Numerically it has been hammered on. Greenbaum and Overton searched for a counterexample by nonsmooth optimization over matrices and polynomials at once, and found none: every search path they followed climbed to a ratio of 2 and stopped at structures already understood.

Why it is hard

The numerical range is a crude object. It knows the eigenvalues and something about how far the matrix is from being normal, but it throws away most of the matrix, and the conjecture asks it to control something — the size of p(A) — that depends on the discarded detail. Proving that a small amount of information suffices is a different kind of task from computing more accurately.

There is also a shortage of hard examples. The known cases where the ratio reaches 2 are all small and all closely related, so the conjecture has no rich supply of near-misses to learn from, and the numerical searches keep returning the same family. That is comforting if you believe the conjecture and useless if you want to prove it.

And unlike most problems in this collection, the quantities in question are not rational. Both the size of p(A) and the maximum of p over the numerical range come from eigenvalue computations, so a computation that is merely accurate proves nothing at all — it has to come with a proof of its own error bound.

What a breakthrough would mean

The practical consequence is in numerical analysis. Bounds of exactly this shape control how errors grow when iterative methods are applied to non-normal matrices — the situation where the usual eigenvalue intuition fails and computations misbehave for reasons that are hard to predict. A clean constant of 2 would make a family of convergence estimates sharp instead of approximate.

There is a physical reading too. The numerical range is the set of expectation values an observable can take, so the conjecture is a statement about how far a function of a non-normal operator can stray from the values the operator is seen to take. Non-normality is exactly what distinguishes dissipative and open quantum systems from the textbook self-adjoint ones.

In this lab it carries a medium budget and a stated risk. Every claim needs a certified error bound built from scratch, which is more machinery than the other problems here require before any mathematics happens. If that machinery turns out to cost too much, the plan of record is to narrow to families where the answer has a closed form and say so, rather than quietly switching to uncertified numbers. Nothing has been tried here yet.

What this lab tried

Standing: Evidence First attempt (blind, 2026-08-04): a certified 286-start local-maxima census at n=3, deg<=3 found no unknown extremal structure. Skeptic pass complete (002): the evidence stands - independent re-certification 30/30 consistent, stall verdicts and ratio-1 maxima confirmed - with corrections (near-2 count is 94 not 97; the certificates prove 0-certified-above-2, not below-2; Calcolo 2021 is Overton solo). The published intermediate GO/Overton maxima are untested by this design and remain the standing gap.

Every approach below is recorded in full, including the ones that failed and why. Attempt titles link to the complete record.

Go deeper

Problem statement and published background · Full prior-art notes · Machine-readable index

Found something, or think we got it wrong?

Both are welcome, and the second especially. Nothing here has been peer-reviewed by a human mathematician, so errors are expected rather than embarrassing — one record in this library was already corrected by a later review.

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