Let \(A=UDU^{-1}\in \mathbb{C}^{n\times n}\) where \[D=\begin{pmatrix}\lambda_1&&\\ &\ddots&\\ &&\lambda_n\end{pmatrix}\] be a diagonalizable matrix with eigenvalues \(\lambda_1,\ldots,\lambda_n.\) The eigenvector condition number of \(A\) is defined as \[\kappa_V(A):=\inf_{A=VDV^{-1}}\|V\|\|V^{-1}\|.\] If \(\lambda_1,\ldots,\lambda_n\) are distinct, then the eigenvalue condition number of \(\lambda_i\) is defined as \[\kappa(\lambda_i):=\|v_i\|\|w_i\|\] where \(v_i\) is the \(i\)-th column of \(U\) and \(w_i^*\) is the \(i\)-th row of \(U^{-1}\) .
In this paper, the authors study the following question posed by E. B. Davies [SIAM J. Matrix Anal. Appl. 29, No. 4, 1051–1064 (2008; Zbl 1157.65024)]: how well can an arbitrary matrix be approximated by one with a small eigenvector condition number?
They give an answer in Theorem 1.1 which is an improvement to a result in [loc. cit.].
Theorem. Suppose that \(A\in \mathbb{C}^{n\times n}\) and \(\delta\in(0,1)\). Then there is a matrix \(E\in \mathbb{C}^{n\times n}\) such that \(\|E\|\le\delta\|A\|\) and \[\kappa_V(A+E)\le 4n^{3/2}\left(1+\frac{1}{\delta}\right).\]
Consequently, they confirme a conjecture given in [loc. cit.].
Conjecture. For every positive integer \(n\) there is a constant \(c_n\) such that for every \(A\in \mathbb{C}^{n\times n}\) with \(\|A\|\le 1\) and \(\varepsilon\in(0,1)\), \[\inf_{E\mathbb{C}^{n\times n}}(\kappa_V(A+E)\varepsilon+\|E\|)\le c_n\sqrt{\varepsilon}.\]
They also state Theorem 1.5 which concerns the eigenvalue condition numbers. Theorem 1.5 shows that adding a small Ginibre perturbation regularizes the eigenvalue condition numbers of any matrix in an averaged sense.
Theorem. Suppose that \(A\in \mathbb{C}^{n\times n}\) with \(\|A\|\le 1\) and \(\delta\in(0,1)\). Let \(G_n\) be a complex Ginibre matrix, and let \(\lambda_1,\ldots,\lambda_n\in C\) be the (random) eigenvalues of \(A+\delta G_n\). Then for every measureable open set \(B\subset \mathbb{C}\), \[\mathbb{E}\sum_{\lambda_i\in B}\kappa(\lambda_i)^2\le \frac{n^2}{\pi\delta^2}\mathrm{Vol}(B).\]
Section 2 is the preparation work for the proofs, given in Section 3. Along the way, they give a proof to Śniady’s comparison theorem (Theorem 2.3) and adopt the technique to prove Theorem 2.4, which is the real version of Theorem 2.3. The details of the proof of Theorem 2.4 are in the appendix. A consequence of Theorem 2.4 is Proposition 2.5, which is a conjecture of A. Sankar et al. [SIAM J. Matrix Anal. Appl. 28, No. 2, 446–476 (2006; Zbl 1179.65033)].
Proposition. Let \(G\) be an \(n\times n\) matrix with i.i.d. real \(N(0,1)\) entries and \(A\) be any \(n\times n\) matrix with real entries. Then \[\mathbb{P}[\sigma_n(A+G)<\varepsilon]\le\varepsilon\sqrt{n}.\]
In Section 4, the authors show that the bound in Theorem 1.1 is the best possible for large \(n\). In Section 5, they pose some questions for further studies.