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numerical-range:examples:ginibre [2013/09/27 14:22]
gawron [Numerical range of random matrices]
numerical-range:examples:ginibre [2018/10/08 08:01] (current)
plewandowska
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 ====== Numerical range of random matrices ====== ====== Numerical range of random matrices ======
-Complex Ginibre matrices $G_N$ of order $N$ with entries $\xi_{ij}$, where + 
-$\mathbb{E} |\xi _{ij}|^2 =1/N$. As we mention in the introduction,​ by the  circular law,+The figures below depict numerical range for large random matrices drawn from different ensembles. 
 +Gray areas denote numerical ranges and red dots denote spectra of matrices. 
 + 
 +Matrices are normalized so that for every matrix $A$ $\Tr(AA^\dagger)=\dim(A)$. 
 + 
 +In the figures below  
 +  * $r(X)= {\rm max}\{|z|: z \in W(X)\}$ denotes the matrix **numerical radius**  
 +  * $\rho(X)= ​ |\lambda_{\max}|$ denotes the matrix **spectral radius**, 
 +where $\lambda_{\rm max}$ is the leading eigenvalue of $X$ 
 +with the largest modulus. 
 + 
 + 
 +===== Definitions ===== 
 +Complex Ginibre matrices $G_d$ of dimention ​$d$ with entries $\xi_{ij}$, where 
 +$\mathbb{E} |\xi _{ij}|^2 =1/d$. As we mention in the introduction,​ by the  circular law,
 the spectrum of $G_N$ is asymptotically contained in the unit disk. the spectrum of $G_N$ is asymptotically contained in the unit disk.
-Note $\mathbb{E} \|G_N\| _{\rm HS}^2=N$.+Note $\mathbb{E} \|G_d\| _{\rm HS}^2=d$.
  
  
-Upper triangular random matrices $T_N$  of order $N$ with entries $T_{ij}=\xi_{ij}$ for $i <j$ +Upper triangular random matrices $T_d$  of dimension ​$d$ with entries $T_{ij}=\xi_{ij}$ for $i <j$ 
-and $T_{ij}=0$ elsewhere, where $\mathbb{E} |\xi _{ij}|^2 =2/(N-1)$.+and $T_{ij}=0$ elsewhere, where $\mathbb{E} |\xi _{ij}|^2 =2/(d-1)$.
 Clearly, all eigenvalues of $T_N$  equal to zero. Clearly, all eigenvalues of $T_N$  equal to zero.
-Note $\mathbb{E} \|T_N\| _{\rm HS}^2=N$.+Note $\mathbb{E} \|T_d\| _{\rm HS}^2=d$.
  
  
-Diagonalized Ginibre matrices, $D_N = Z G_N Z^{-1}$ ​ of order $N$, so that +Diagonalized Ginibre matrices, $D_d = Z G_d Z^{-1}$ ​ of dimension ​$d$, so that 
-$D_{kl}=\lambda_k \delta_{kl}$ where $\lambda_k$,​ $k=1,​\dots, ​N$, +$D_{kl}=\lambda_k \delta_{kl}$ where $\lambda_k$,​ $k=1,​\dots, ​d$, 
-denote complex eigenvalues of  $G_N$.+denote complex eigenvalues of  $G_d$.
 In order to ensure the uniqueness of the probability distribution on In order to ensure the uniqueness of the probability distribution on
 diagonal matrices, diagonal matrices,
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 one gets the average squared eigenvalue of the complex Ginibre matrix, one gets the average squared eigenvalue of the complex Ginibre matrix,
 $\langle |\lambda|^2\rangle =\int_{0}^1 2x^3 dx=1/2$. Thus, $\langle |\lambda|^2\rangle =\int_{0}^1 2x^3 dx=1/2$. Thus,
-$\mathbb{E} \|D_N\| _{\rm HS}^2=N/2$.+$\mathbb{E} \|D_d\| _{\rm HS}^2=d/2$.
  
-Diagonal unitary matrices $U_N$  of order $N$ with entries $U_{kl}=\exp(i \phi_k) \delta_{kl}$,​+Diagonal unitary matrices $U_d$  of order $d$ with entries $U_{kl}=\exp(i \phi_k) \delta_{kl}$,​
 where $\phi_k$ are independent uniformly distributed on $[0, 2 \pi)$ real random where $\phi_k$ are independent uniformly distributed on $[0, 2 \pi)$ real random
 variables. variables.
  
-Matrices are normalized so that for every matrix $A$ $\Tr(AA^\dagger)=\dim(A)$. +Examples: 
- +$$ 
-Red dots denote spectra of matrices.+G_4 = \left[\begin{array}{cccc} 
 +                  \xi  & \xi & \xi & \xi \\ 
 +                  \xi  & \xi & \xi & \xi \\ 
 +                  \xi  & \xi & \xi & \xi \\ 
 +                  \xi  & \xi & \xi & \xi \\ 
 +\end{array}\right],​ \ 
 +T_4 = \left[\begin{array}{cccc} 
 +                    ​0 ​ & \xi & \xi &  \xi \\ 
 +                    ​0 ​ &  0  & \xi &  \xi \\ 
 +                    0  &  0  &  0  &  \xi \\ 
 +                    0  &  0  &  0  &   ​0 ​ \\ 
 +\end{array}\right],​ \ 
 + ​D_4 ​ = \left[\begin{array}{cccc} 
 +                    \lambda_1 & 0  &   ​0 ​ &   0 \\ 
 +                    0  & \lambda_2 &   ​0 ​ &   0 \\ 
 +                    0  &  0  &  \lambda_3 &   0 \\ 
 +                    0  &  0  &   ​0 ​ &  \lambda_4 \\ 
 +\end{array}\right] ​. 
 +\label{struct} 
 +$$
  
-In the figures below  
-$r(X)= {\rm max}\{|z|: z \in W(X)\}$, 
-denotes the //numerical radius// 
-and  
-$\rho(X)= ​ |\lambda_{\max}|$ 
-denotes the //spectral radius//, 
-where $\lambda_{\rm max}$ is the leading eigenvalue of $X$ 
-with the largest modulus, 
 ==== Matrices drawn from Ginibre ensemble ==== ==== Matrices drawn from Ginibre ensemble ====
  
 <​html><​center></​html>​ <​html><​center></​html>​
-Matrix ​$G$, dimension ​10 ^+A matrix ​$G_{10}$ ^
 | {{ :​numerical-range:​g_10.png?​nolink |}} | | {{ :​numerical-range:​g_10.png?​nolink |}} |
 <​html></​center></​html>​ <​html></​center></​html>​
  
 <​html><​center></​html>​ <​html><​center></​html>​
-Matrix ​$G$, dimension ​100 ^+A matrix ​$G_{100}$ ^
 | {{ :​numerical-range:​g_100.png?​nolink |}} | | {{ :​numerical-range:​g_100.png?​nolink |}} |
 <​html></​center></​html>​ <​html></​center></​html>​
  
 <​html><​center></​html>​ <​html><​center></​html>​
-Matrix ​$G$, dimension ​1000 ^+A matrix ​$G_{1000}$ ^
 | {{ :​numerical-range:​g_1000.png?​nolink |}} | | {{ :​numerical-range:​g_1000.png?​nolink |}} |
 <​html></​center></​html>​ <​html></​center></​html>​
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 ===== Diagonal matrices ===== ===== Diagonal matrices =====
 <​html><​center></​html>​ <​html><​center></​html>​
-Matrix ​$D$, dimension ​10 ^+A matrix ​$D_{10}$ ^
 | {{ :​numerical-range:​d_10.png?​nolink |}} | | {{ :​numerical-range:​d_10.png?​nolink |}} |
 <​html></​center></​html>​ <​html></​center></​html>​
  
 <​html><​center></​html>​ <​html><​center></​html>​
-Matrix ​$D$, dimension ​100 ^+A matrix ​$D_{100}$ ^
 | {{ :​numerical-range:​d_100.png?​nolink |}} | | {{ :​numerical-range:​d_100.png?​nolink |}} |
 <​html></​center></​html>​ <​html></​center></​html>​
  
 <​html><​center></​html>​ <​html><​center></​html>​
-Matrix ​$D$, dimension ​1000 ^+A matrix ​$D_{1000}$ ^
 | {{ :​numerical-range:​d_1000.png?​nolink |}} | | {{ :​numerical-range:​d_1000.png?​nolink |}} |
 <​html></​center></​html>​ <​html></​center></​html>​
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 <​html><​center></​html>​ <​html><​center></​html>​
-Matrix ​$T$, dimension ​10 ^+A matrix ​$T_{10}$ ^
 | {{ :​numerical-range:​t_10.png?​nolink |}} | | {{ :​numerical-range:​t_10.png?​nolink |}} |
 <​html></​center></​html>​ <​html></​center></​html>​
  
 <​html><​center></​html>​ <​html><​center></​html>​
-Matrix ​$T$, dimension ​100 ^+A matrix ​$T_{100}$ ^
 | {{ :​numerical-range:​t_100.png?​nolink |}} | | {{ :​numerical-range:​t_100.png?​nolink |}} |
 <​html></​center></​html>​ <​html></​center></​html>​
  
 <​html><​center></​html>​ <​html><​center></​html>​
-Matrix ​$T$, dimension ​1000 ^+A matrix ​$T_{1000}$ ^
 | {{ :​numerical-range:​t_1000.png?​nolink |}} | | {{ :​numerical-range:​t_1000.png?​nolink |}} |
 <​html></​center></​html>​ <​html></​center></​html>​
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 ==== Combinations ==== ==== Combinations ====
 <​html><​center></​html>​ <​html><​center></​html>​
-Matrix ​$D+1/\sqrt{2}T$, dimension ​1000 ^+A matrix ​$D_{1000}+1/\sqrt{2}T_{1000}$ ^
 | {{ :​numerical-range:​d_tsq2_1000.png?​nolink |}} | | {{ :​numerical-range:​d_tsq2_1000.png?​nolink |}} |
 <​html></​center></​html>​ <​html></​center></​html>​
  
 <​html><​center></​html>​ <​html><​center></​html>​
-Matrix ​$U+T$, dimension ​1000 ^+A matrix ​$U_{1000}+T_{1000}$ ^
 | {{ :​numerical-range:​u_t_1000.png?​nolink |}} | | {{ :​numerical-range:​u_t_1000.png?​nolink |}} |
 <​html></​center></​html>​ <​html></​center></​html>​
  
numerical-range/examples/ginibre.1380291751.txt.gz · Last modified: 2013/09/27 14:22 by gawron