What is the Relation between Eigenvalues & Singular Values? Mario Kieburg

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1 CRC701 What is the Relation between Eigenvalues & Singular Values? Mario Kieburg Fakultät für Physik; Universität Bielefeld Fakultät für Physik; Universität Duisburg-Essen Hausdorff Center for Mathematics (Bonn) January 12th, 2016

2 In Collaboration with: Holger Kösters Faculty of Mathematics, Bielefeld University Kieburg, Kösters: arxiv: [math.ca]

3 The Main Question in this Talk Re Im z C eigenvalue of X C n n : det(x z 1) = 0 λ 0 singular value of X C n n : det(x X λ 2 1) = 0

4 The Main Question in this Talk Re Im When I give you the singular values of a matrix, what are its eigenvalues? What is the relation vice versa?

5 Outline of this Talk What is known? What isn t known? The Answer for Bi-Unitarily Invariant Ensembles! Example: Polynomial Ensembles Idea & Results

6 In the most General Case Assume ordering: determinant, the only equality: n det X X = z j 2 = eigenvalues squared singular values z 1... z n and a 1... a n Weyl s inequalities ( 49), k = 1,..., n: k z j 2 k Horn s inequalities ( 54), k = 1,..., n: k z j 2 k a j n We have only inequalities! a j a j

7 Normal Matrices X is normal : [X, X ] = 0 equalities: z j 2 = a j, j = 1,..., n special case, Hermitian matrices: z j R a j = z 2 j special case, unitary matrices: z j = e ıϕ j S 1 C a j = 1 Inequalities become equalities!

8 Normal Random Matrices A particular model (e.g. Chau, Zaboronsky ( 98); Teodorescu et al. ( 05); Bleher, Kuijlaars ( 12)): g = k zk distributed by f G (g)dg = f ev (z)dzd k d k: Haar measure of U (n) = K joint density of ev s f ev (z) n (z) 2 n ω( z j 2 ) χ(z j ) 2 χ: analytic function n (z) = i<j (z j z i ) χ(z) = 1, jpdf of squared sv s f sv (λ) Perm[a j 1 i ω(a i )] Re Im No level repulsion for singular values!

9 Bi-Unitarily Invariant Random Matrices f G (g) = f G (k 1 gk 2 ), for all g Gl (n) = G and k 1, k 2 U (n) = K Schur decomposition: g = k ztk with unitary matrix: k U (n) = K unitriangular matrix: t T complex diagonal matrix: z [Gl (1)] n = Z joint density of eigenvalues n f ev (z) n (z) 2 z j 2n 2j f G (zt)dt singular value decomposition: g g = k ak with unitary matrix: k U (n) = K positive diagonal matrix: a R n + = A joint density of squared singular values f sv (a) n (a) 2 f G ( a) Ev s and sv s usually exhibit level repulsion! T

10 Bi-Unitarily Invariant Random Matrices joint density of eigenvalues n f ev (z) n (z) 2 z j 2n 2j T joint density of squared singular values f sv (a) n (a) 2 f G ( a) f G (zt)dt What is the relation between f ev and f sv?

11 Bi-Unitarily Invariant Random Matrices (at large matrix dimension n) single ring theorem (Feinberg, Zee ( 97); Guionnet et al. ( 11)) bi-unitary invariance + some conditions connected support for radii Haagerup-Larson theorem (Haagerup, Larsen ( 00), Haagerup, Schultz ( 07)) bi-unitary invariance bijection between level densities ρ ev and ρ sv Ρ sv Λ

12 Bi-Unitarily Invariant Random Matrices (Product of infinitely many matrices) spectral statistics of g 1 g M with g j Bi-Unitarily invariant random matrix and M at finite n for particular Meijer G-ensembles: Akemann, Kieburg, Burda ( 14); Ipsen ( 15) singular values and radii of eigenvalues become deterministic at the same positions

13 Bi-Unitarily Invariant Random Matrices What is the relation between f ev and f sv at finite n and M? Is there a bijection between f ev and f sv?

14 Example: Polynomial Ensembles Definition: Let w 0,..., w n 1 and ω some functions with suitable integrability and differentiability conditions. (a) f sv is polynomial ensemble (Kuijlaaars et al. ( 14/ 15)) : f sv (a) n (a) det[w j 1 (a i )] (b) f sv is polynomial ensemble of derivative type (Kieburg, Kösters ( 16)) : w j 1 (a) = ( a a ) j 1 ω(a) (c) f sv is Meijer G-ensemble (Kieburg, Kösters ( 16)) : ω is Meijer G-function

15 Example: Polynomial Ensembles Some polynomial ensembles of derivative type: Laguerre (χgaussian, Ginibre, Wishart) ensemble: ω(a) = a ν e a Jacobi (truncated unitary) ensemble: ω(a) = a ν (1 a) µ 1 Θ(1 a) Cauchy-Lorentz ensemble: ω(a) = a ν (1 + a) ν µ 1 products of random matrices: ω(a) =Meijer G-function ΡΛ Gauss Jacobi Lorentz Muttalib-Borodin of Laguerre-type Λ (a) ω(a) = a ν e αaθ generating n (a θ ) (b) θ 0: ω(a) = a ν e α (ln a) 2 generating n (ln a) (c) θ : ω(a = 1 + a /θ) = e νa e αea generating n (e a ) works also for Jacobi-type or even Cauchy-Lorentz-type

16 Example: Polynomial Ensembles Laguerre: f sv (a) = 2 n(a)det ν ae tr a n (a) det[( a i ai ) j 1 ai ν e a i ], f ev (z) = N n (z) 2 z i 2ν e z i 2 Jacobi: f sv (a) = 2 n(a)det ν a det(1 n a) µ n n (a) det[( a i ai ) j 1 ai ν (1 a i) µ 1 ], f ev (z) = n n (z) 2 z i 2ν (1 z i 2 ) µ 1 similar for other known Meijer G-ensembles Does this simple relation hold for other ensembles as well?

17 The Idea jpdf s of bi-unitarily invariant random matrices L 1,K (G) I Ω jpdf s of positive definite matrices L 1,K (Ω) I A T H S L 1,EV (Z ) jpdf s of ev s I Z L H (A) jpdf s of radii (almost) jpdf s of sv s L 1,SV (A) M ML H (A) Mellin transform of L H (A) This diagram is commutative! All maps are linear and bijective!

18 jpdf s of bi-unitarily invariant random matrices L 1,K (G) The Idea I Ω jpdf s of positive definite matrices L 1,K (Ω) I A T H S L 1,EV (Z ) jpdf s of ev s I Z L H (A) jpdf s of radii (almost) jpdf s of sv s L 1,SV (A) M ML H (A) Mellin transform of L H (A) g g in positive definite Hermitian matrices y Ω = Gl (n)/u (n): I Ω f G (y) f G ( y), I 1 Ω f Ω(g) f Ω (g g) positive definite diagonal matrices A in Ω: I A f Ω (a) n (a) 2 f Ω (a), I 1 A f sv(y) f sv (λ(y))/ n (λ(y)) 2 (λ(y): ev s of y Ω and squared sv s of g G)

19 jpdf s of bi-unitarily invariant random matrices L 1,K (G) The Idea I Ω jpdf s of positive definite matrices L 1,K (Ω) I A T H S L 1,EV (Z ) jpdf s of ev s I Z L H (A) jpdf s of radii (almost) jpdf s of sv s L 1,SV (A) M ML H (A) Mellin transform of L H (A) positive definite diagonal matrices (squared radii) A in Z : I Z f A (z) n (z) 2 det n 1 z f A ( z ), I 1 Z f fz ( aφ)d Φ Z (a) [ ] Perm (Φ: complex phases of the eigenvalues) multivariate Mellin transform: a (2j+n 3)/2 i Mf A (s) Perm[a s j 1 i ]f A (a)da, M 1 ([Mf A ]; a) Perm[a s i ]Mf A (s)ds

20 jpdf s of bi-unitarily invariant random matrices L 1,K (G) The Idea I Ω jpdf s of positive definite matrices L 1,K (Ω) I A T H S L 1,EV (Z ) jpdf s of ev s I Z L H (A) jpdf s of radii (almost) n T f G (z) n (z) 2 z j 2n 2j jpdf s of sv s L 1,SV (A) M ML H (A) Mellin transform of L H (A) T f G (zt)dt This is the crucial operator we are looking for! Bijectivity? Explicit Representation?

21 jpdf s of bi-unitarily invariant random matrices L 1,K (G) The Idea I Ω jpdf s of positive definite matrices L 1,K (Ω) I A T H S L 1,EV (Z ) jpdf s of ev s I Z L H (A) jpdf s of radii (almost) jpdf s of sv s L 1,SV (A) M ML H (A) Mellin transform of L H (A) Harish-transform (Harish-Chandra ( 58)) ) Hf Ω (a) T f Ω(t at)dt ( n a(n 2j+1)/2 j factorization (Kieburg, Kösters ( 16)): H = I 1 Z T I 1 Ω

22 jpdf s of bi-unitarily invariant random matrices L 1,K (G) The Idea I Ω jpdf s of positive definite matrices L 1,K (Ω) I A T H S L 1,EV (Z ) jpdf s of ev s I Z L 1,H (A) jpdf s of radii (almost) spherical-transform (Harish-Chandra ( 58)) Sf Ω (s) f Ω (y)ϕ(y, s)dy/det n y spherical function (Gelfand, Naĭmark ( 50)) ϕ(y, s) det[(λ i(y)) s j +(n 1)/2 ] n (s) n (λ(y)) S is invertible (Harish-Chandra ( 58)) factorization (Harish-Chandra ( 58)): S = MH jpdf s of sv s L 1,SV (A) M ML 1,H (A) Mellin transform of L 1,H (A)

23 variant densities L (G). The diagram is indeed richer in its interpretation than ly the relation between the eigenvalues and singular value because L 1,K ( ) are e K-invariant densities on the space of positive definite Hermitian matrices and H (A) are, apart from a trivial factor (det a) (n Theorem: SEV-Transform R 1)/2, the joint densities of the radii the eigenvalues. Only the image ML 1,H (A) of the Mellin transform of L 1,H (A) auxiliary. In subsections 3.2 and 3.3 we consider two direct applications of this new map. hese applications are to polynomial ensembles and to a particular set of non-biitarily invariant ensembles, respectively. The SEV-(singular value-eigenvalue) transform 1. Mapping between Singular Value and Eigenvalue Statistics. Our first sult is the bijective map between the space L 1,SV (A) of the densities for the uared singular values and the space L 1,EV (Z) of the densities for the eigenvalues bi-unitarily invariant densities on G. heorem 3.1 (Map between fsv and fev). The map R = T I 1 Ω I 1 A : jpdf s of sv s L 1,SV (A) R = TI is bijective 1 and has the explicit representation: I 1 A : L1,SV (A)! L 1,EV (Z) (3.2) om the joint densities of the singular values L 1,SV (A) =IAI L 1,K (G) to the int densities of the eigenvalues L 1,EV (Z) =T L 1,K (G) induced by a bi-unitarily variant signed densities L 1,K (G) is bijective and has the explicit integral reprentation fev(z) = RfSV(z) (3.3) Q n 1 j=0 = j! Z (n!) 2 n n(z) 2 lim 1( (s ı% 0 ))Perm 2(c+ısc) zb!0 R n b,c=1,...,n Z A fsv(a) det[ac+ısc b ]b,c=1,...,n n(% 0 + ıs) n(a) ny Y daj n dsj aj 2 jpdf s of ev s L 1,EV (Z ) SINGULAR VALUE AND EIGENVALUE STATISTICS 20 with fsv 2 L 1,SV (A) and % 0 = diag(% 0 1,...,%0 n), % 0 j =(2j + n 1)/2 with j = 1,...,n,and fsv(a)=r 1 fev(a) (3.4) n Z = (n!) 2 Q n 1 j=0 j! n(a)lim n s ı% 0 + ı n 1 11n!0 n(% 0 + ıs) R n 2 Z det[a c ısc b ]b,c=1,...,n A Z fev( p ny a 0 ) [U(1)] n Perm[a 0c+ısc b ]b,c=1,...,n Perm[a 0c 1 b ]b,c=1,...,n d'j 2 Y n da 0 j a 0 j with fev 2 L 1,EV (Z) for its inverse. The diagonal matrix of phases is = diag(e ı'1,...,e ı'n ) 2 [U(1)] n and the regularizing functions is We call R the SEV-transform. ny 2l cos sj l(s) = Q l k=1 ( 2 4s 2 j /(2k, l 2 N. 1)2 ) (3.5) Note that the integral representation (3.3) is indeed a simplification compared to Eq. (2.7) where we have to integrate over T. The number of integration variables is reduced from n(n 1) for the integral over T to 2n for the operator R. Moreover we have an explicit representation of the inverse R 1 which was not known before not to mention that it was known to be invertible. Please, don t try to read this!

24 Corollary: Polynomial Ensembles of Derivative Type f sv is polynomial ensemble of derivative type: f sv (a) = R 1 f ev (a) n (a) det[( a i ai ) j 1 ω(a i )] + bi-unitary invariance of g = k 1 ak2 G f ev has the form: n f ev (z) = Rf sv (z) n (z) 2 ω( z j 2 ) Note, the arrow works in both directions!

25 Corollary: Implications for the Spectral Statistics Determinantal point process: joint density of squared singular values [ ] n 1 f sv (a) = det K sv (a i, a j ) = p l (a i )q l (a j ) joint density of eigenvalues [ ] n 1 f ev (z) = det K ev (z i, z j ) = ω( z i 2 )ω( z j 2 (z i z j ) l ) Relations: polynomials: p l (a) = 1 2 weights: q l (a) = 1 2 l! ( a) l 0 dr π π π kernel: K sv (a 1, a 2 ) = 1 a2 2 n a 2 dr 0 π l=0 l=0 c l dϕ(ae ıϕ r) l K ev ( r, re ıϕ ) dϕe ılϕ K ev ( a, ae ıϕ ) π π dϕ(a 2 a 1 e ıϕ ) n 1 K ev ( r, re ıϕ )

26 Corollary: Singular Values times Unitary Matrix positive definite diagonal matrix a A distributed by f sv L 1,SV (A) considering either of the random matrices: (a) g = k 1 ak 2, with unitary matrices k 1, k 2 K Haar distributed (b) g = ak or g = ka, with unitary matrix k K Haar distributed (c) g = k 0 ak or g = kak 0, with unitary matrices k 0 K fixed and k K Haar distributed joint density of the eigenvalues of g is f ev = Rf sv We do not need full bi-unitary invariance!

27 Further Results (a) extends to signed densities and distributions (b) generalization to deformations breaking the bi-unitary invariance f G (g) = f (K ) G (g)d G(g) with f (K ) (K ) G (g) = f G (k 1gk 2 ) and D G (g) = D G (g 1 0 gg 0) for all k 1, k 2 K = U (n) and g 0, g G = Gl (n) joint densities: f ev (z) = D G (z)t f (K ) G (z) f sv (a) = n (a) 2 f (K ) G ( a) K D G ( ak)d k (c) products of polynomial ensembles of derivative type semi-group (d) semi-group action on polynomial ensembles transformation law of kernels ála Claeys, Kuilaars, Wang ( 15)

28 Recent developments in RMT RMT enters a new Era! image from de.best-wallpaper.net

29 Announcement! Organizers: Peter Forrester Mario Kieburg Roland Speicher When: August 22nd - 26th 2016 after summer school Where: ZiF next to Bielefeld University Homepage: Thank you for your attention!