Pseudospectral Shattering, the Sign Function, and Diagonalization in Nearly Matrix Multiplication Time

We exhibit a randomized algorithm which given a square $ntimes n$ complex matrix $A$ with $|A| le 1$ and $delta>0$, computes with high probability invertible $V$ and diagonal $D$ such that $$|A-VDV^{-1}|le delta $$ and $|V||V^{-1}| le O(n^{2.5}/delta)$ in $O(T_{MM}>(n)log^2(n/delta))$ arithmetic operations on a floating point machine with $O(log^4(n/delta)log n)$ bits of precision. Here $T_{MM}>(n)$ is the number of arithmetic operations required to multiply two $ntimes n$ complex matrices numerically stably, with $T_{MM},,(n)=O(n^{omega+eta}>>)$ for every $eta>0$, where $omega$ is the exponent of matrix multiplication. The algorithm is a variant of the spectral bisection algorithm in numerical linear algebra (Beavers and Denman, 1974). This running time is optimal up to polylogarithmic factors, in the sense that verifying that a given similarity diagonalizes a matrix requires at least matrix multiplication time. It significantly improves best previously provable running times of $O(n^{10}/delta^2)$ arithmetic operations for diagonalization of general matrices (Armentano et al., 2018), and (w.r.t. dependence on $n$) $O(n^3)$ arithmetic operations for Hermitian matrices (Parlett, 1998). The proof rests on two new ingredients. (1) We show that adding a small complex Gaussian perturbation to any matrix splits its pseudospectrum into $n$ small well-separated components. This implies that the eigenvalues of the perturbation have a large minimum gap, a property of independent interest in random matrix theory. (2) We rigorously analyze Roberts Newton iteration method for computing the matrix sign function in finite arithmetic, itself an open problem in numerical analysis since at least 1986. This is achieved by controlling the evolution the iterates pseudospectra using a carefully chosen sequence of shrinking contour integrals in the complex plane.