Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble
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Annales Henri Poincar´ e
Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble Yan V. Fyodorov and Wojciech Tarnowski Abstract. We study the distribution of the eigenvalue condition numbers κi = (l∗i li )(r∗i ri ) associated with real eigenvalues λi of partially asymmetric N × N random matrices from the real Elliptic Gaussian ensemble. The large values of κi signal the non-orthogonality of the (bi-orthogonal) set of left li and right ri eigenvectors and enhanced sensitivity of the associated eigenvalues against perturbations of the matrix entries. We derive the general finite N expression for the joint density function (JDF) PN (z, t) of t = κ2i − 1 and λi taking value z, and investigate its several scaling regimes in the limit N → ∞. When the degree √ of asymmetry is fixed as N → ∞, the number of real eigenvalues is O( N ), and in the the specbulk of the real spectrum ti = O(N ), while on approaching √ tral edges the non-orthogonality is weaker: ti = O( N ). In both cases the corresponding JDFs, after appropriate rescaling, coincide with those found in the earlier studied case of fully asymmetric (Ginibre) matrices. A different regime of weak asymmetry arises when a finite fraction of N eigenvalues remain real as N → ∞. In such a regime eigenvectors are weakly non-orthogonal, t = O(1), and we derive the associated JDF, finding that the characteristic tail P(z, t) ∼ t−2 survives for arbitrary weak asymmetry. As such, it is the most robust feature of the condition number density for real eigenvalues of asymmetric matrices. Mathematics Subject Classification. 60B20 Random matrices. Keywords. Bi-orthogonal eigenvectors, Eigenvalue condition numbers, Weak non-Hermiticity.
1. Introduction A (real-valued) square matrix X is asymmetric if it is different from its transpose X T , and non-normal if XX T = X T X. Generically, asymmetric matrices
Y. V. Fyodorov, W. Tarnowski
Ann. Henri Poincar´e
are non-normal, and their eigenvalues are much more sensitive to the perturbations of the matrix entries than for their symmetric (hence selfadjoint and normal) counterparts. It is well known that non-normality may raise serious issues when calculating the spectra of such matrices numerically: keeping a fixed precision of calculations might not be sufficient, as some eigenvalues can be ‘ill-conditioned’. To be more specific, we assume that X can be diagonalized (which for random matrices happens with probability one). Then to each eigenvalue λi , real or complex (in the latter case being always accompanied by its complex conjugate partner λi ) correspond two sets of eigenvectors, left li and right ri which can always be chosen to be bi-orthogonal: l∗i rj = δij , where l∗i := lTi stands for Hermitian conjugation. The corresponding eigenproblems are Xri = λi ri and X T li = λi li . Consider now a matrix X = X + P , where the second term represents an error one makes by storing the matrix entries with a finite precision, with > 0 controlling the magnitude of the error and P reflecting the matrix structure of the
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