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Letbe an × matrix with real i.i.d. (0, 1/ ) entries, let be a real × matrix with ‖ ‖ ≤ 1, and let ∈ (0, 1). We show that with probability 0.99, + has all of its eigenvalue condition numbers bounded by 5/2 / 3/2 and eigenvector condition number bounded by 3 / 3/2 . Furthermore, weshow that for any > 0, the probability that + has two eigenvalues within distance at most of each other is 4 1/3 / 5/2 . In fact, we show the above statements hold in the more general setting of non-Gaussian perturbations with real, independent, absolutely continuous entries with a finite moment assumption and appropriate normalization. This extends the previous work [BKMS19] which proved an eigenvector condition number bound of 3/2 / for the simpler case of complex i.i.d. Gaussian matrix perturbations. The case of real perturbations introduces several challenges stemming from the weaker anticoncentration properties of real vs. complex random variables. A key ingredient in our proof is new lower tail bounds on the small singular values of the complex shifts − ( + ) which recover the tail behavior of the complex Ginibre ensemble when ℑ ≠ 0. This yields sharp control on the area of the pseudospectrum Λ ( + ) in terms of the pseudospectral parameter > 0, which is sufficient to bound the overlaps and eigenvector condition number via a limiting argument.
Letbe an × matrix with real i.i.d. (0, 1/ ) entries, let be a real × matrix with ‖ ‖ ≤ 1, and let ∈ (0, 1). We show that with probability 0.99, + has all of its eigenvalue condition numbers bounded by 5/2 / 3/2 and eigenvector condition number bounded by 3 / 3/2 . Furthermore, weshow that for any > 0, the probability that + has two eigenvalues within distance at most of each other is 4 1/3 / 5/2 . In fact, we show the above statements hold in the more general setting of non-Gaussian perturbations with real, independent, absolutely continuous entries with a finite moment assumption and appropriate normalization. This extends the previous work [BKMS19] which proved an eigenvector condition number bound of 3/2 / for the simpler case of complex i.i.d. Gaussian matrix perturbations. The case of real perturbations introduces several challenges stemming from the weaker anticoncentration properties of real vs. complex random variables. A key ingredient in our proof is new lower tail bounds on the small singular values of the complex shifts − ( + ) which recover the tail behavior of the complex Ginibre ensemble when ℑ ≠ 0. This yields sharp control on the area of the pseudospectrum Λ ( + ) in terms of the pseudospectral parameter > 0, which is sufficient to bound the overlaps and eigenvector condition number via a limiting argument.
We exhibit a randomized algorithm which, given a square matrix $$A\in \mathbb {C}^{n\times n}$$ A ∈ C n × n with $$\Vert A\Vert \le 1$$ ‖ A ‖ ≤ 1 and $$\delta >0$$ δ > 0 , computes with high probability an invertible V and diagonal D such that $$ \Vert A-VDV^{-1}\Vert \le \delta $$ ‖ A - V D V - 1 ‖ ≤ δ using $$O(T_\mathsf {MM}(n)\log ^2(n/\delta ))$$ O ( T MM ( n ) log 2 ( n / δ ) ) arithmetic operations, in finite arithmetic with $$O(\log ^4(n/\delta )\log n)$$ O ( log 4 ( n / δ ) log n ) bits of precision. The computed similarity V additionally satisfies $$\Vert V\Vert \Vert V^{-1}\Vert \le O(n^{2.5}/\delta )$$ ‖ V ‖ ‖ V - 1 ‖ ≤ O ( n 2.5 / δ ) . Here $$T_\mathsf {MM}(n)$$ T MM ( n ) is the number of arithmetic operations required to multiply two $$n\times n$$ n × n complex matrices numerically stably, known to satisfy $$T_\mathsf {MM}(n)=O(n^{\omega +\eta })$$ T MM ( n ) = O ( n ω + η ) for every $$\eta >0$$ η > 0 where $$\omega $$ ω is the exponent of matrix multiplication (Demmel et al. in Numer Math 108(1):59–91, 2007). The algorithm is a variant of the spectral bisection algorithm in numerical linear algebra (Beavers Jr. and Denman in Numer Math 21(1-2):143–169, 1974) with a crucial Gaussian perturbation preprocessing step. Our result significantly improves the previously best-known provable running times of $$O(n^{10}/\delta ^2)$$ O ( n 10 / δ 2 ) arithmetic operations for diagonalization of general matrices (Armentano et al. in J Eur Math Soc 20(6):1375–1437, 2018) and (with regard to the dependence on n) $$O(n^3)$$ O ( n 3 ) arithmetic operations for Hermitian matrices (Dekker and Traub in Linear Algebra Appl 4:137–154, 1971). It is the first algorithm to achieve nearly matrix multiplication time for diagonalization in any model of computation (real arithmetic, rational arithmetic, or finite arithmetic), thereby matching the complexity of other dense linear algebra operations such as inversion and QR factorization up to polylogarithmic factors. 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. In particular, this implies that the eigenvalues of the perturbed matrix have a large minimum gap, a property of independent interest in random matrix theory. (2) We give a rigorous analysis of Roberts’ Newton iteration method (Roberts in Int J Control 32(4):677–687, 1980) for computing the sign function of a matrix in finite arithmetic, itself an open problem in numerical analysis since at least 1986.
Let A$A$ be an n×n$n\times n$ real matrix, and let M$M$ be an n×n$n\times n$ random matrix whose entries are independent and identically distributed sub‐Gaussian random variables with mean 0 and variance 1. We make two contributions to the study of sn(A+M)$s_n(A+M)$, the smallest singular value of A+M$A+M$. (1)We show that for all ε⩾0$\epsilon \geqslant 0$, double-struckP[snfalse(A+Mfalse)⩽ε]=O(εn)+2e−normalΩfalse(nfalse),\begin{equation*} \mathbb {P}[s_n(A + M) \leqslant \epsilon ] = O(\epsilon \sqrt {n}) + 2e^{-\Omega (n)}, \end{equation*}provided only that A$A$ has normalΩfalse(nfalse)$\Omega (n)$ singular values which are Ofalse(nfalse)$O(\sqrt {n})$. This extends a well‐known result of Rudelson and Vershynin, which requires all singular values of A$A$ to be Ofalse(nfalse)$O(\sqrt {n})$. (2)We show that any bound of the form sup∥A∥⩽nC1double-struckP[snfalse(A+Mfalse)⩽n−C3]⩽n−C2\begin{equation*} \sup _{\Vert A\Vert \leqslant n^{C_1}}\mathbb {P}[s_n(A+M)\leqslant n^{-C_3}] \leqslant n^{-C_2} \end{equation*}must have C3=normalΩ(C1C2)$C_3 = \Omega (C_1 \sqrt {C_2})$. This complements a result of Tao and Vu, who proved such a bound with C3=O(C1C2+C1+1)$C_3 = O(C_1C_2 + C_1 + 1)$, and counters their speculation of possibly taking C3=O(C1+C2)$C_3 = O(C_1 + C_2)$.
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