A function of a matrix is at most twice its size on the numerical range
The Crouzeix conjecture of 2004 holds with the sharp constant 2, for every matrix and every bounded operator.
The result in plain terms
Numerical computing applies functions to matrices all the time. An iterative solver applies a polynomial of the matrix to a starting error. A time-stepping scheme applies a rational function of it once per step. The practical question is how large the result can be, because that size is the error or the growth.
For a symmetric matrix the eigenvalues answer the question: the size of $f(A)$ is the largest value of $|f|$ on the eigenvalues. For a matrix that is not symmetric, the eigenvalues can mislead badly. A matrix whose eigenvalues are all zero need not be zero, and a system whose eigenvalues all say "decay" can grow a thousandfold before it decays.
The numerical range $W(A)$ is a safer region to look at. It is the set of numbers $x^*Ax$ over all unit vectors $x$. It is a convex region of the complex plane, it contains the eigenvalues, and its outline can be computed from a few dozen symmetric eigenvalue problems. The theorem says
$$\lVert f(A)\rVert \;\le\; 2\,\max_{z\in W(A)} |f(z)|$$
for every square matrix $A$ of any size and every polynomial $f$, and by extension every function analytic near $W(A)$. A function of a matrix is never more than twice as large as the function is on the numerical range. The same bound holds when the coefficients of $f$ are themselves matrices, and for bounded operators in infinite dimensions.
The constant 2 cannot be lowered. The matrix $\begin{pmatrix}0&2\\0&0\end{pmatrix}$ has norm 2, its numerical range is the unit disc, and $f(z)=z$ is at most 1 there.
What was known
Michel Crouzeix conjectured the constant 2 in 2004 and proved the bound with 11.08 in place of 2 in 2007. In 2017 Crouzeix and César Palencia brought the constant down to $1+\sqrt2\approx 2.414$. The gap between 2.414 and 2 stayed open, with the conjecture confirmed only for special classes of matrices.
Why it is usable today
The theorem is an inequality. There is no algorithm to implement and no running time to worry about. Every published estimate that carries the factor $1+\sqrt2$ now holds with 2, a tightening of 17 per cent.
- Iterative solvers. After $k$ steps of GMRES on $Ax=b$, the residual satisfies $\lVert r_k\rVert/\lVert r_0\rVert \le 2\min_p \max_{z\in W(A)}|p(z)|$, the minimum being over polynomials of degree $k$ with $p(0)=1$. The right-hand side is a problem about one complex variable on a convex region.
- Functions of matrices. If a polynomial or rational function $p$ approximates $f$ to within $\delta$ on $W(A)$, then $p(A)$ approximates $f(A)$ to within $2\delta$. Any scalar approximation result on a convex region becomes a matrix result at the cost of a factor of two. Rational approximations to the matrix exponential are analysed this way.
- Time stepping. A scheme that advances a linear system by $u_{n+1}=r(hA)u_n$ is stable when $|r(hz)|\le 1$ on $W(A)$, with the solution bounded by twice its starting size for all time.
What it does not give
The gain over the 2017 bound is a constant factor of 1.2, and no computation becomes faster. Bounds built on the numerical range are weak when the numerical range is much larger than the region where the eigenvalues sit, and the GMRES bound says nothing when $W(A)$ contains zero. Those limits belong to the numerical range and are unchanged.
Scores and sources
Applied score 5 of 5, difficulty score 4 of 5. The problem dates from about 2004. A main theorem of this family is in OpenAI's Lean formalization catalogue. The scores take the theorem as stated.