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Carlo Beenakker
Carlo Beenakker
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Indeed, if $A$ is non-Hermitian you cannot use the inverse $(A-z)^{-1}$ to study eigenvalues near a complex number $z$. To apply resolvent techniques to a non-Hermitian matrix $A$ you need to first symmetrize it, $$H(z)=\begin{pmatrix}0&A-z\\ (A-z)^\ast&0 \end{pmatrix}.$$ Then Girko's formula$^\ast$ relates the eigenvalues $\lambda$ of $A$ to the resolvent of the Hermitian matrix $H(z)$ on the imaginary axis. A review of this technique, and how it works in the limit that the dimension $n$ of $A$ goes to infinity, is Fluctuations in the spectrum of non-Hermitian i.i.d. matrices (2022).

$$\sum_{\lambda}f(\lambda)=-\frac{1}{4\pi}\int\Delta f(z)\int_0^\infty\operatorname{Im}\operatorname{Tr}[H(z)-i\eta]^{-1}d\eta\, d^2 z.\qquad(\ast)$$$^\ast$ V.L. Girko, Theory of Probability and its Applications 29, 694 (1985). $$\sum_{\lambda}f(\lambda)=-\frac{1}{4\pi}\int\Delta f(z)\int_0^\infty\operatorname{Im}\operatorname{Tr}[H(z)-i\eta]^{-1}d\eta\, d^2 z.$$

Indeed, if $A$ is non-Hermitian you cannot use the inverse $(A-z)^{-1}$ to study eigenvalues near a complex number $z$. To apply resolvent techniques to a non-Hermitian matrix $A$ you need to first symmetrize it, $$H(z)=\begin{pmatrix}0&A-z\\ (A-z)^\ast&0 \end{pmatrix}.$$ Then Girko's formula$^\ast$ relates the eigenvalues $\lambda$ of $A$ to the resolvent of the Hermitian matrix $H(z)$ on the imaginary axis. A review of this technique, and how it works in the limit that the dimension $n$ of $A$ goes to infinity, is Fluctuations in the spectrum of non-Hermitian i.i.d. matrices (2022).

$$\sum_{\lambda}f(\lambda)=-\frac{1}{4\pi}\int\Delta f(z)\int_0^\infty\operatorname{Im}\operatorname{Tr}[H(z)-i\eta]^{-1}d\eta\, d^2 z.\qquad(\ast)$$

Indeed, if $A$ is non-Hermitian you cannot use the inverse $(A-z)^{-1}$ to study eigenvalues near a complex number $z$. To apply resolvent techniques to a non-Hermitian matrix $A$ you need to first symmetrize it, $$H(z)=\begin{pmatrix}0&A-z\\ (A-z)^\ast&0 \end{pmatrix}.$$ Then Girko's formula$^\ast$ relates the eigenvalues $\lambda$ of $A$ to the resolvent of the Hermitian matrix $H(z)$ on the imaginary axis. A review of this technique, and how it works in the limit that the dimension $n$ of $A$ goes to infinity, is Fluctuations in the spectrum of non-Hermitian i.i.d. matrices (2022).

$^\ast$ V.L. Girko, Theory of Probability and its Applications 29, 694 (1985). $$\sum_{\lambda}f(\lambda)=-\frac{1}{4\pi}\int\Delta f(z)\int_0^\infty\operatorname{Im}\operatorname{Tr}[H(z)-i\eta]^{-1}d\eta\, d^2 z.$$
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Carlo Beenakker
Carlo Beenakker
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Indeed, if $A$ is non-Hermitian you cannot use the inverse $(A-z)^{-1}$ to study eigenvalues near a complex number $z$. To apply resolvent techniques to a non-Hermitian matrix $A$ you need to first symmetrize it, $$H(z)=\begin{pmatrix}0&A-z\\ (A-z)^\ast&0 \end{pmatrix}.$$ Then Girko's formula$^\ast$ relates the eigenvalues $\lambda$ of $A$ to the resolvent of the Hermitian matrix $H(z)$ on the imaginary axis. A review of this technique, and how it works in the limit that the dimension $n$ of $A$ goes to infinity, is Fluctuations in the spectrum of non-Hermitian i.i.d. matrices (2022).

$$\sum_{\lambda}f(\lambda)=-\frac{1}{4\pi}\int\Delta f(z)\int_0^\infty\operatorname{Im}\operatorname{Tr}[H(z)-i\eta]^{-1}d\eta\, d^2 z.\qquad(\ast)$$

Indeed, if $A$ is non-Hermitian you cannot use the inverse $(A-z)^{-1}$ to study eigenvalues near a complex number $z$. To apply resolvent techniques to a non-Hermitian matrix $A$ you need to first symmetrize it, $$H(z)=\begin{pmatrix}0&A-z\\ (A-z)^\ast&0 \end{pmatrix}.$$ Then Girko's formula relates the eigenvalues $\lambda$ of $A$ to the resolvent of the Hermitian matrix $H(z)$. A review of this technique, and how it works in the limit that the dimension $n$ of $A$ goes to infinity, is Fluctuations in the spectrum of non-Hermitian i.i.d. matrices (2022).

Indeed, if $A$ is non-Hermitian you cannot use the inverse $(A-z)^{-1}$ to study eigenvalues near a complex number $z$. To apply resolvent techniques to a non-Hermitian matrix $A$ you need to first symmetrize it, $$H(z)=\begin{pmatrix}0&A-z\\ (A-z)^\ast&0 \end{pmatrix}.$$ Then Girko's formula$^\ast$ relates the eigenvalues $\lambda$ of $A$ to the resolvent of the Hermitian matrix $H(z)$ on the imaginary axis. A review of this technique, and how it works in the limit that the dimension $n$ of $A$ goes to infinity, is Fluctuations in the spectrum of non-Hermitian i.i.d. matrices (2022).

$$\sum_{\lambda}f(\lambda)=-\frac{1}{4\pi}\int\Delta f(z)\int_0^\infty\operatorname{Im}\operatorname{Tr}[H(z)-i\eta]^{-1}d\eta\, d^2 z.\qquad(\ast)$$
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Carlo Beenakker
Carlo Beenakker
  • 188.2k
  • 18
  • 448
  • 651

Indeed, if $A$ is non-Hermitian you cannot use the inverse $(A-z)^{-1}$ to study eigenvalues near a complex number $z$. To apply resolvent techniques to a non-Hermitian matrix $A$ you need to first symmetrize it, $$H(z)=\begin{pmatrix}0&A-z\\ (A-z)^\ast&0 \end{pmatrix}.$$ Then Girko's formula relates the eigenvalues $\lambda$ of $A$ to the resolvent of the Hermitian matrix $H(z)$. A review of this technique, and how it works in the limit that the dimension $n$ of $A$ goes to infinity, is Fluctuations in the spectrum of non-Hermitian i.i.d. matrices (2022).