SupelecRandom Matrix TheoryforWireless CommunicationsM´erouane Debbahhttp://[email protected], 2008PresentationDefinitions, Probability and convergence measures1Circularly SymmetricWe say that a random vector x with complex entries is circularly symmetric whenE[xxT] = 0 even when E[xxH] 6= 0.2EigenvaluesP. Lancaster, ”Theory of Matrices”, Academic Press, NY, 1969.Any complex matrix H can be written:H = URUHwhere U is unitary (U−1= UH) and R is an upper triangular matrix.The eigenvalues of H are the principal diagonal elements of R. If H is hermitian(H = H∗), then R is diagonal and the eigenvalues are real.3Rayleigh QuotientDefinition. The extreme eigenvalues of a Hermitian matrix H can be characterized interms of the Rayleigh quotient R(x) defined by:R(x) =xHHxxHxLemma. Given a Hermitian matrix H, let λminand λmaxdenote respectively theminimum and maximum eigenvalues of H. Thenλmin= minxR(x)λmax= maxxR(x)4Norm of a matrixDefinition. A norm T (H) on the space of N × N matrices satisfies the followingproperties:(1) T (H) ≥ 0 with equality if and only if H = 0 is the all zero matrix.(2) For any two matrices H1and H2,T (H1+ H2) ≤ T (H1) + T (H2)(3) For any scalar α and matrix H,T (αH) =| α | T (H)5Hilbert-Schmidt normFor an N × N complex matrix W = (wij), the Hilbert-Schmidt norm (or Schur norm orEuclidean norm or Frobenius norm) of W is defined as:|| W ||=sXij| wij|26Useful Inequalities1. | Trace(BC) |≤|| B |||| C ||2. If U is a unitary matrix, then for any C of the same order,|| CU ||=|| UC ||=|| C ||3. For a Hermitian matrix B with eigenvalues λ1, ..., λN, and any C,max(|| BC ||, || CB ||) ≤ max1≤i≤N| λi||| C ||7Useful Inequalities1. For a rectangular matrix A and B of the same size,rank(A + B) ≤ rank(A) + rank(B)2. For rectangular matrices A and B in which AB is defined:rank(AB) ≤ min(rank(A), rank(B))3. For Hermitian N × N matrices A and B,NXi=1(λAi− λBi)2≤ Trace(A − B)24. For a rectangular A,rank(A) ≤ the number of non-zero entries ofA8Matrix Inversion LemmaLet A, C and C−1+ DA−1B be non-singular square matrices, then(A + BCD)−1= A−1B³C−1+ DA−1B´−1DA−19Matrix Inversion in Block FormLet a (M + N) × (M + N) matrix W be partitioned into a block form:W =·A BC D¸A is M × M, B is M × N, C is N × M and D is N × N . In this case, we have:W−1=·(A − BD−1C)−1−A−1B(D − CA−1B)−1−D−1C(A − BD−1C)−1(D − CA−1B)−1¸10Unitarily Invariant MatricesAn N × N self-adjoint random matrix W is called unitarily invariant if the probabilitymeasure of W as a random matrix is equal to that of the matrix VWVHfor any unitaryconstant matrix V.Remark. Hermitian matrices are also called self-adjoint since if W is Hermitian, then inthe usual inner product of CN, we have: (u, Wv) = (Wu, v) for all (u, v) ∈ CN.11Weierstrass theoremTheorem. If F (x) is a continuous function on [a, b], there exists a sequence ofpolynomials pn(x) such thatlimn→∞pn(x) = F (x)uniformly on [a, b].In other words, any real function defined on a real interval can be approximated arbitrarilyclosely and uniformly by a polynomial.12LpnormConsider I to be a closed real interval. Denote B(I) the Borel set of I and µ theLebesgue measure of any set of B(I). The measurable space of X with measure µ isrepresented by (X, B(X), µ).Definition. The space Lpon the measurable space (R, B(R), µ) is defined as follows:Lp= {f :Z| f(x) |pdµ(x) < ∞}where p is such that 1 ≤ p < ∞.Definition. Let f ∈ Lp, then we define the norm in Lpas the functional || . ||Lp: Lp→ R,defined by|| f ||Lp=µZ∞−∞| f(x) |pdx¶1p13Some Basic Notions of Measure theoryRoughly speaking, a measure is a function that assigns a number (size, volume..) tosubsets of a given set.When is it useful? When one wants to carry out integration over arbitrary sets ratherthan on an interval.14σ-algebraDefinition. A σ-algebra over a set X is a collection Σ of subsets of X that is closedunder complementation and countable unions of its members, in other words:• If A is in Σ then so is the complement of A.• The union of countably many sets in Σ is also in Σ.Elements of the σ-algebra are called measurable sets and the pair (X, Σ) is called ameasurable space.15Definitions of a MeasureDefinition. A measure µ is a function defined on a σ-algebra Σ over a set S takingvalues in the interval [0, ∞] with the following properties:• The empty set has measure zero:µ(®) = 0• Countable additivity: If E1, E2, E3, .. is a countable sequence of pairwise disjoint setsin Σ, then:µ(∞[i=1Ei) =∞Xi=1µ(Ei).The triple (S, Σ, µ) is then called a measure space and the members of Σ are calledmeasurable sets.16The Lebesgue MeasureHenri L´eon Lebesgue, 1875-1941The Lebesgue measure is the standard way of assigning a number (length, area orvolume) to subsets of Euclidean space. It is used to define the Lebesgues integration.Henri Lebesgue described his measure in 1901, followed the next year by his descriptionof the Lebesgue integral. Both were published as part of his dissertation in 1902.17Density and distributionDefinition. Let f ∈ L1. We say that f is a density if and only if f (x) ≥ 0 for almost allx ∈ R and || f ||L1= 1.Definition. A right continuous positive and non decreasing function F on R withlimx→−∞F (x) = 0 and limx→∞F (x) = 1 is called a distribution.18Convergence in law{xn} represents an infinite sequence of random variables x1, x2, ... defined in the sameprobability space and taking values on either the real or complex fieldsConvergence in Law. Let Fn(x), F (x) denote the probability distribution functions of xnand x respectively. We say that xnconverges in law to x if limn→∞Fn(x) = F (x) atevery continuity point of F .Note: Convergence in law is also known as the weak convergence or convergence indistribution.Remark: x is a random variable defined on the same probability space.19Convergence in probabilityConvergence in probability. We say that xnconverges in probability to x iflimn→∞Pr[| xn− x |< ²] = 1 ∀² > 0Note: Convergence in probability involves the joint distribution of (xn, x) whereasweak convergence concerns only the (marginal) distribution of xn.20Convergence in the rthmeanConvergence in the rthmean. We say that xnconverges in the rthmean to x iflimn→∞E[| xn− x |r] = 0The most commonly used for of convergence in the rthmean is the convergence in themean square sense.It can