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• Linear algebra exercises 0. Prove that that for any n × n matrix M there exists a factorization of the form: M = U DV ∗ where U and V are n × n unitary matrices, and D is diagonal. 1. Show that a positive-defenite hermitian n × n matrix A has a unique positivedefinite square root B. (That is B 2 = A. Positive-definite means that for all nonzero complex vectors v we have v ∗ Av > 0.) 2. Let A and B be two symmetric matrices with common eigenbasis, show that AB is symmetric. 3. Let λ1 ≥ · · · ≥ λn denote the eigenvalues of n × n matrix. Let A and B be two Hermitian matrices. Prove that λ1 (A + B) ≤ λ1 (A) + λ1 (B). (Hint: show that λ(A) = sup|v|=1 v ∗ Av) 4. Show that for any Hermitian n × n matrix A we have λi (A) = sup inf v ∗ Av dim(V )=i v∈V,|v|=1 where V ranges over all subspaces of Cn with the indicated dimension. 5. Show that for any Hermitian n × n matrix A with top left n − 1 × n − 1 minor An−1 the following enequality holds λi+1 ≤ λ1 (An−1 ) ≤ λi (An ) for all 1 ≤ i ≤ n − 1 6. Show that for any two Hermitian n × n matrices A and B the following enequality holds n X |λi (A) − λi (B)|2 ≤ tr(A − B)2 i=1 (Hint: observe that this inequality means that the maximum over matrices A and B with a given spectrum of the right hand side is achieved when the two matrices have the same basis of eigenvectors and more precisely the k-th eigenvector correspond to the k-th largest eigenvalues of the matrices. This fact can be shown by induction over the dimension n of the matrices). • Catalan numbers Prove that the number of triangulations of a convex (n + 2)-gon is equal to the Catalan number cn • Relaxation of the assumptions on the moments of the entries Let AN be a sequence of Hermitian matrices with iid entries such that (1) E[AN ij ] = 0, 1 ≤ i, j ≤ N 1 2 (2) E[(AN ij ) ] = N 1 2 Then 1 lim Tr (AN )k = N →∞ N ( 0, Ck , 2 if k is odd, otherwise, where the convergence holds in expectation and almost surely. (Hint: fix a costant C and consider the approximation of AN by AˆN ij = q N √ √ AN ij 1 N |AN |≤C − E[Aij 1 N |AN |≤C ] ij √ E[AN ij 1 N |AN ij |≤C ij − √ E[AN ]] ij 1 N |AN ij |≤C . Use the enequality from the exercise 6. ) • Band matrices Take XN,L an Hermitian N ×N matrix such that XN,L (ij) vanishes for |i−j| ≥ L and otherwise XN,L (ij), i ≤ j ≤ i + L are independent, equidistributed√random variables which are centered and with covariance (2L)−1 . Assume that the LXN,L have uniformly bounded moments, independent of L, N Show that 1 k Tr(XN,L )] = Ck/2 L→∞ N →∞ N where Ck/2 = 0 if k is odd and otherwise is the Catalan number. Hint: expand the expectation in terms of the entries, as in the course consider the graph associated with the indices and show that the main contribution is given by indices such that these graphs are trees. lim lim E[ • Wishart matrices We consider a N × M matrix XN,M with independent and equidistributed real entries which are centered and with variance N −1 . We assume that √ sup max E[( N XN,M (ij))2k ] < ∞ ∀k ≥ 0 N i,j We let YN,M be the M × M matrix ∗ YN,M = XN,M XN,M ∗ (ij) = XN,M (ji). with XN,M (1) Assume M stays finite while N goes to infinity. Show that YN,M converges almost surely towards the identity in the set of M × M matrices. (Hint: use the law of large numbers) (2) Assume that M goes to infinity while M/N goes to zero. Show that for all bounded continuous function f , M 1 X f (λi ) − f (1) M i=1 goes to zero in probability. Hint: show that if IM (ij) = 1i=j , 1 ≤ i, j ≤ M , 1 tr(YN,M − IM )2 ] = 0. M (3) Assume M, N goes to infinity so that M/N goes to one. -Show that for all k ≥ 0, lim E[ M →∞ lim E[ N →∞ 1 k Tr(YN,M )] = Ck N 3 with Ck the Catalan number, that is the number of non-crossing pair partition of 2k points. Hint: generalize the proof of Wigner’s theorem: say why the limit is the same. -Deduce that for all bounded continuous function √ Z 4 M 1 X 1 4−x f (x) √ lim E[ f (λi )] = dx N →∞ M i=1 2π 0 x (4) Assume M, N goes to infinity so that M/N goes to α ∈ (0, 1). We want to show that r−1 lim E[ N →∞ X 1 r Tr(YN,M )] = αk Ck,r M k=0 where Ck,r counts certain subsets of the non-crossing pair partitions of 2r points. More precisely, consider 2r points on the line and checkerboard color the intervals between points by red or blue, starting with blue the interval connected to −∞. Color in blue the faces between the real line and the non-crossing partitions with first interval on the real line which is blue in a blue interval. Ck,r is the number of non-crossing pair partitions of 2r points with k + 1 blue faces. Hint: -Write the expectation X 1 ∗ E[XN,M (i1 i2 )XN,M (i2 i3 ) · · · XN,M (i2r i1 )] mN (r) := E[ Tr(YN,M )r ] = M 1≤i ≤M 2j+1 1≤i2j ≤N and consider the graph Gi = (Vi , Ei ) with vertices Vi = (i` )1≤`≤2r and edges Ei = (ik ik+1 , k ≤ 2r − 1, (i2r i1 )) as in the course. Show that the leading contribution to mN (r) corresponds to the case where the skeleton of Gi is a tree. Consider the coloring of the associated non-crossing partition defined above and show that each non crossing pair-partitions with k + 1, 1 ≤ k ≤ r − 1, blue faces will correspond to M k+1 N r−k choices of indices. Conclude. ∗ be a symmetric N × N • CLT for Heavy tailed matrices Let XN = XN matrix with i.i.d centered entries such that for all k ≥ 0 lim N E[XN (ij)2k ] = Dk N →∞ P with k≥2 Dk > 0. Show that 1 k k σk := lim E[ (tr(XN ) − E[tr(XN )])2 ] = N →∞ N X X G,G0 ∈Tk P ∈Pk (G) E∩E 0 6=∅ P 0 ∈Pk (G0 ) Y DmP (e)+mP 0 (e) e∈E∪E 0 where the sum runs over -G, G0 are rooted trees with at most k/2 edges (that is live in Tk ), who share an edge [that is E ∩ E 0 6= ∅] (Trees here have edges with multiplicity 1). -P, P 0 are closed paths on G and G0 respectively, with length equal to k, starting at the root. 4 -2(mP (e) + mP 0 (e)) is the number of times that the path P or P 0 went through the edge e (counted as undirected). Compare to the case of Wigner matrices whose entries have moments of order k √ −k of order N .