Products of Rectangular Gaussian Matrices

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1 Products of Rectangular Gaussian Matrices Jesper R. Ipsen Department of Physics Bielefeld University in collaboration with Gernot Akemann Mario Kieburg Summer School on Randomness in Physics & Mathematics August 2013

2 Why Products of Random Matrices? Transport in disordered materials Wireless telecommunications Quantum entanglement Finance Etc.

3 A Single Matrix

4 What is a Ginibre Matrix? Non-Hermitian Gaussian Matrix (Ginibre Matrix) X 1 : Rectangular N 1 N 0 matrix Entries [X 1 ] ab : Independent Complex Gaussians Notation: ν 1 N 1 N 0 0 dμ(x 1 ) = e Tr X 1X 1 dx1 Ginibre (1965), etc.

5 What is an Induced Ginibre Matrix? Induced Ginibre Matrix X1 X 1 = U 1 0 X 1 : N 1 N 0 Ginibre matrix U 1 U(N 1 )/[U(N 0 ) U(ν 1 )] X 1 : N 0 N 0 Induced Ginibre matrix dμ Ind (X 1 ) = det ν 1 [X 1 X 1 ] e Tr X 1X 1 dx1 Fischmann, Bruzda, Khoruzhenko, Sommers & Życzkowski (2012)

6 Eigenvalues Schur Decomposition X 1 = U(Z + T)U X 1 : (Induced) Ginibre matrix U U(N 0 )/U(1) N 0 Z = diag(z 1,..., z N0 ): Eigenvalues T: Strictly upper triangular matrix

7 Eigenvalues Joint Probability Density Function P jpdf (Z) N0 z l 2ν 1 e z l 2 det (zi ) j 1 det (z i,j a,b a )b 1 l=1 Correlation Functions R ν k (z N 0! 1,..., z k ) = (N 0 k)! N 0 l=k+1 d 2 z l P jpdf (z 1,..., z N0 ) C

8 Eigenvalue Distribution m z Radial 0.4 Re z Radial density Numerical Radial cut Edges r in = ν 1 r out = N 1 r = z Numerical Analytical Macroscopic

9 Singular Values Singular Value Decomposition X 1 = UΣV X 1 : Ginibre matrix U, V: Unitary matrices Σ = diag(σ 1,..., σ N0 ): Singular values

10 Singular Values Joint Probability Density Function P jpdf (Σ) N0 l=1 σ l det i,j σ 2(j 1) i det a,b σ 2(ν 1+b 1) a e σ2 a Correlation Functions R ν k (σ N 0! 1,..., σ k ) = (N 0 k)! N 0 l=k+1 0 dσ l P jpdf (σ 1,..., σ N0 )

11 Singular Values Numerical Analytical Macroscopic (Marčenko Pastur) 1 Density Singular value

12 Products of Matrices

13 Products Ginibre Matrices The Product Matrix Y M = X M X 2 X 1 Y M : N M N 0 matrix X i : N m N m 1 Ginibre Matrix Notation: ν m N m N 0 0 (N 0 is smallest matrix size)

14 Finding Eigen- & Singular Values Two Main Points from the Derivation 1. Rectangular matrices Induced matrices 2. Integration over Meijer G-functions

15 Parameterization The Product Matrix Y M = X M X 2 X 1 Parameterization X1 X 1 = U 1 0 X 1 : N 0 N 0 matrix U 1 U(N 1 ) [U(N 0 ) U(N 1 N 0 )]

16 Parameterization The Product Matrix X1 Y M = X M X 3 X 2 U 1 0 Parameterization X2 A X 2 2 U 1 = U 2 0 B 2 X 2 : N 0 N 0 matrix A 2 : N 0 (N 1 N 0 ) matrix B 2 : (N 2 N 0 ) (N 1 N 0 ) matrix U 2 U(N 2 ) [U(N 0 ) U(N 2 N 0 )]

17 Parameterization The Product Matrix X2 A Y M = X M X 4 X 2 X1 3 U 2 0 B 2 0 Parameterization X 3 : N 0 N 0 matrix X3 A X 3 3 U 2 = U 3 0 B 3 A 3 : N 0 (N 2 N 0 ) matrix B 3 : (N 3 N 0 ) (N 2 N 0 ) matrix U 3 U(N 3 ) [U(N 0 ) U(N 3 N 0 )]

18 Parameterization Final Parameterization XM A Y M X2 A 2 X1 YM M = U M = U 0 B M 0 B 2 0 M 0 Y M = X M X M 1 X 1 : N 0 N 0 matrix X m : Induced Ginibre matrices

19 Finding Eigen- & Singular values Two Main Points from the Derivation 1. Rectangular matrices Induced matrices 2. Integration over Meijer G-functions

20 Meijer G-function Definition G m, n a1,..., a p p, q b 1,..., b q z = m 1 du z u i=1 Γ[b i u] n i=1 Γ[1 a i + u] 2πi p i=n+1 Γ[a i u] q i=m+1 Γ[1 b i + u] Path m[] Re[]

21 Meijer G-function A Simple Example G 1, 0 0, 1 b 1 z = 1 2πi path du z u Γ[b 1 u] = R k = z b 1 e z k=0 m[] Re[] b 1

22 Joint Probability Density Functions JPDF for Eigenvalues N0 P EV jpdf (Z) l=1 z l 2ν 1 e z l 2 det i,j (zi ) j 1 det (z a,b a )b 1 JPDF for Singular Values N0 P SV jpdf (Σ) l=1 σ l det i,j σ 2(j 1) i det a,b σ 2(ν 1+b 1) a e σ2 a

23 Joint Probability Density Functions JPDF for Eigenvalues N0 P EV jpdf (Z) l=1 G 1, 0 0, 1 ν1 zl 2 det i,j (zi ) j 1 det (z a,b a )b 1 JPDF for Singular Values N0 P SV jpdf (Σ) l=1 σ l det i,j σ 2(j 1) i det a,b G 1, 0 0, 1 ν 1 +b 1 σ 2 a

24 Meijer G-function Integration formula 0 dt e t t b 0 1 G m, n p, q a1,..., a p s b 1,..., b q t = G m+1, n p, q+1 a1,..., a p b 0, b 1,..., b q s

25 Joint Probability Density Functions JPDF for Eigenvalues N0 P EV jpdf (Z) l=1 G M, 0 0, M ν M,..., ν 1 zl 2 det i,j (zi ) j 1 det (z a,b a )b 1 JPDF for Singular Values N0 P SV jpdf (Σ) l=1 σ l det i,j det a,b σ 2(j 1) i G M, 0 0, M ν M,..., ν 2, ν 1 +b 1 σ 2 a

26 Eigenvalue Distribution Numerical Analytical Macroscopic r in = ν 1 ν M, r out = N 1 N M Radial density r = z

27 Singular Value Distribution Numerical Analytical Macroscopic Burda, Jarosz, Livan, Nowak & Swiech (2010) 1 Density Singular value

28 Correlation Functions Correlations for Eigenvalues R EV k (z 1,..., z k ) = k l=1 G M, 0 0, M ν M,..., ν 1 zl 2 det 1 a,b k N0 1 n=0 z a z b M m=1 (n + ν m)! Correlations for Singular Values R SV k (s 1,..., s k ) = det 1 a,b k N0 1 n=0 G 1, 0 1, M+1 G M, 1 1, M+1 n+1 0, ν M,..., ν sa 1 n ν M,..., ν 1,0 s b