Comptes rendus de l Academie bulgare des Sciences, Tome 59, 4, 2006, p POSITIVE DEFINITE RANDOM MATRICES. Evelina Veleva

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1 Comtes rendus de l Academie bulgare des ciences Tome POITIVE DEFINITE RANDOM MATRICE Evelina Veleva Abstract: The aer begins with necessary and suicient conditions or ositive deiniteness o a matrix These are used or obtaining an equivalent by distribution resentation o elements o a Wishart matrix by algebraic unctions o indeendent random variables The next section studies when sets o samle correlation coeicients o the multivariate normal (Gaussian) distribution are deendent indeendent or conditionally indeendent ome joint densities are obtained The aer ends with an algorithm or generating uniormly distributed ositive deinite matrices with reliminary ixed diagonal elements Key words: ositive deinite matrix Wishart distribution multivariate normal (Gaussian) distribution samle correlation coeicients generating random matrices Mathematics ubject Classiication: 6H Necessary and suicient conditions or ositive deiniteness o a matrix Deinition uose A is a real symmetric matrix I or any non-zero vector x we have that x A x> then A is a ositive deinite matrix (here and subsequently x is the transose o x ) Let A = { a } n i j= be a real matrix and let N = { n} For any nonemty α β N let A ( α β ) be the submatrix o A obtained by deleting rom A the rows indexed by α and the columns indexed by β To simliy notation we use A ( α ) instead o A ( α α) We will write the comlement o α in N as α c For any integers i and j i j n we will denote by A ( α ) and A ( α) the block matrices A( α) c c A( α) A( α{ j} ) A( α) A( α{ j} ) = c A( α) ( α{} i ) a = c A A ( α{} i ) Theorem Let A be a real symmetric matrix o size n and let i j be ixed integers such that i< j n The matrix A is ositive deinite i and only i the matrices A({ i}) and A({ j}) are both ositive deinite and the element a satisies the inequalities A A A A + A A < a < A({ }) A({ }) ({ i j}) ({ i}) ({ j}) ({ i j}) ({ i}) ({ j})

2 where denotes determinant o a matrix Theorem A real symmetric matrix A = { a } n = is ositive deinite i and only i its elements satisy the ollowing conditions: i j a > i= n; ii a a < a < a a i= n; ii i ii A A A ({ i n}) ({ i n}) ii ({ i n}) jj A({ i n}) < a A + A A < A({ i n}) ({ i n}) ({ i n}) ii ({ i n}) jj i= n j = i+ n The roos o Theorems and will aear in [] Wishart random matrices Let X Xn be indeendent Xi ~ N ( Σ) with n ie Xi has a - dimensional multivariate normal distribution with mean vector a zero vector and covariance matrix Σ Σ > and let W = XiX i We say that W has a central Wishart distribution with n degrees o reedom and covariance matrix Σ and write W ~ W ( n Σ) It is easy to see that that W is a matrix and that W > From this deinition it is aarent E W = n Σ () AWA ~ W ( n AΣA ) q where A is q o rank q It is known that or an arbitrary ositive deinite matrix Σ there exist a matrix B such that Σ = BB Consequently rom () it ollows that i W~ W ( n I) where I is the identity matrix o size then BWB ~ W ( n Σ ) Thereore rom now on we consider the case Σ = I Ignatov and Nikolova in [] consider the distribution Ψ ( n ) with joint density () where C ( n / ) D I E

3 -C is a constant; - D is the matrix x x x x D = ; x x - I E is the indicator o the set E containing all oints ( x i< j ) in the real sace R such that the matrix D is ositive deinite ( )/ They have roved the ollowing Proosition: Proosition Let τ τ τ be indeendent and identically χ - distributed random variables with n degrees o reedom and let random variables ν i < j have distribution ( n ) Let us assume that the set { τ τ τ } is indeendent o the set Ψ { ν i< j } Then the matrix below V has the Wishart distribution W ( n I ) ν ν τ τ ν ν V= τ τ ν ν Using the same variable transormation as in the roo o this Proosition it can be shown that the oosite statement is also true o the density () gives actually the joint density o the entries o the samle correlation matrix in the case Σ = I The next Theorem gives an equivalent by distribution resentation o the random variables ν i< j having distribution Ψ ( n ) by unctions o indeendent random variables This can be used or generating Ψ ( n ) distributed random matrices and consequently Wishart ones (see [3]) Theorem 3 Let the random variables η i< j be mutually indeendent uose that η η be identically distributed Ψ( n i + ) or i= The ii+ random variables i ν i< j deined by νi = η i i= ν i = η η i + η i ( η )( η i) i= 3 3

4 i i t ν = η ( ηqi )( ηq j ) + ηtiηtj ( ηqi )( ηqj ) 3 i< j q= t= q= have distribution Ψ ( n ) To generate random Ψ( n ) matrices using Theorem 3 we have to know how to generate random variables with distribution Ψ ( k) or k For = the density () o the distribution Ψ( n ) has the orm n 3 C( y ) y ( ) and it is known as the Pearson robability distribution o second tye Random variables with this distribution can be easily generated (see [4] 48) using the quotient o the dierence and the sum o two gamma - distributed random variables 3 Marginal densities o the distribution Ψ ( n ) In recent years the Wishart distribution and distributions derived rom the Wishart one have received a lot o attention because o their use in grahical Gaussian models The essence o grahical models in multivariate analysis is to identiy indeendences and conditional indeendences between various grous o variables Emirical correlation matrices are o great imortance or risk management and asset allocation The study o correlation matrices is one o the cornerstone o Markowitz s theory o otimal ortolios Using Theorems and it is ossible to get some marginal densities o the distribution Ψ( n ) and to investigate when sets o samle correlation coeicients o the multivariate normal (Gaussian) distribution are deendent indeendent or conditionally indeendent Let ν i< j be random variables with distribution Ψ ( n ) Denote by V the set o random variables V = { ν i< j } In [5] we have roved the ollowing Theorem: Theorem 4 Every random variables rom the set V are deendent Let us introduce the notation or the joint density o the random variables rom a set ; i =Φ ie the set is emty we deine = Theorem 5 Let r and q be arbitrary integers such that r and q Let us denote by and the sets = { ν i< j r} and = { ν q+ i< j } Then 4

5 (3) = The roo will aear in [6] We denote by and the densities o / / and conditioned on the random variables rom the set I we divide the two sides o equality (3) by we get the relation / (4) = / / / According to the deinition o conditional indeendence given in [7] equality (4) shows that the random variables rom the sets and are indeendent conditionally on the random variables rom the set Theorem 6 Let q qk r r k be integers such that = q q qk and r r r = Let us denote by the set = { v q i< j r } l = k Then k k l k = 3 k l l l k Let = { ν s= k} be an arbitrary subset o the set o random variables V To i j s s this subset we can attach a grah G ( ) with nodes { } and k undirected edges {{ i j } s= k} s s Theorem 7 Let and be two subsets o the set V and and G be their corresonding grahs Let us denote by G K the set o the numbers o the nodes which are vertex o an edge rom Analogically or the grah G let us denote the corresonding G set by and let K = K K I the set K contains at most one element then K = ie the set and are indeendent I the set K contains at least two elements and or every two elements r and q rom K r < q the random variable ν rq belongs simultaneously to and then = ie the sets and are indeendent conditioned on the set 5

6 Theorem 8 Let be a subset o the set V and G be the corresonding grah Let us denote by the subset o containing all variables which edges in the grah G are + art o a closed ath (circuit) I indeendent is the set = \ then the sets and are + The roos o Theorems 6-8 will aear in [8] In [9] we have roved the ollowing two Theorems: Theorem 9 Let r and q be arbitrary integers such that r < q The random variables rom the set = V \{ ν rq } have joint density unction o the orm where ( n + ) Γ Γ ( rr ; qq ; ) C D D n + Γ D rqrq ; ( n ) ( n + ) - is the indicator o the set E containing all oints in I E R ( )/ D rr ; D qq ; matrices and are both ositive deinite; - Γ() is the well-known Gamma unction I E + such that the Theorem Let r be an arbitrary integer such that r The random variables rom the set = { ν i< j i r j r } have joint density unction o the orm = n Γ n n n + Γ Γ Γ Γ R( )( )/ D rr ; ( )( ) or all oints in or which the matrix is ositive deinite 4 Uniormly distributed ositive deinite matrices The ositive deiniteness o a matrix is a necessary condition or alying many numerical algorithms The diagonal elements o the matrix have oten seciied signiicance The correctness o such numerical algorithm can be roven i we are able to choose a ositive deinite matrix at random with uniorm distribution The sace o all ositive deinite matrices is however a cone and consequently a uniorm distribution cannot be deined over the whole cone because it has ininite volume This enorces to introduce additional restrictions on the matrices which together with the ositive deiniteness reduce our choice within a set with inite volume An algorithm or generating uniormly distributed ositive deinite matrices with ixed diagonal elements is given below I the user does not D rr ; n 6

7 want to ix concrete diagonal elements o the matrix in advance he has though to assign bounds or each diagonal element For instance he can choose all diagonal elements to be in the interval () Then a uniorm random number in the chosen bounds has to be generated or every diagonal element The described below algorithm allows the user a random choice among all ositive deinite matrices with concrete diagonal elements that are either ixed in advance or randomly generated within the chosen bounds From now on we assume that a ann are the diagonal elements o the matrix chosen in accordance with users reerences They have to be ositive so that to exist at least one ositive deinite matrix with such diagonal elements (see Theorem ) Theorem Let the random variables η i< j be mutually indeendent uose that η η be identically distributed Ψ ( n i+ ) or i= ii+ Consider the random variables ν = i ν i< j deined by νi = ηi a aii i= i r i = aa ii jj ηη ri rj ( ηqi )( ηq j ) + η ( ηqi )( ηq j ) r= q= q= i= j = i+ The joint density unction o the random variables ν i< j is o the orm: where - C is the constant ν < ( x i< j ) = CI i j E R ( )/ C = + Γ 3 Γ Γ Γ Γ ( ) ( a a a ) - I E is the indicator o the set E consisting o all oints ( x i< j ) in or which the matrix a x x n x a xn (5) A = xn xn ann is ositive deinite ; 7

8 Theorem gives the ollowing algorithm or generating uniormly distributed ositive deinite matrices: ) Generate ( ) / random numbers y i< j so that comes rom the distribution Ψ( n i+ ) such that ) In order to reduce calculations comute the auxiliary quantities z i j = j i j z y ( y ) ( y ) with y = 3) Calculate the desired matrix (5) with x = aa ( zz + z z + + zz) ii jj i j i j ii ii y The obtained ormulas show that it is ossible to create a rogram which generates without dialog with the user a matrix U with units on the main diagonal This is done by stes )-3) substituting a = = a nn = Then the user according to his reerences orms a diagonal matrix D = diag( a ann ) and the desired matrix A is A= DUD The details will aear in [] Reerence: [] VELEVA E Linear Algebra and Al (submitted) [] IGNATOV T G A D NIKOLOVA Annuaire Univ oia Fac o Econ & Buss Admin 4 v [3] VELEVA E Annuaire Univ oia Fac o Econ & Buss Admin 7 v [4] DEVROYE L Non-Uniorm Random Variate Generation New York ringer-verlag [5] VELEVA E T IGNATOV Advanced tudies in Contemorary Mathematics () [6] VELEVA E Math and Education in Math 35(6) 35-3 [7] DAWID A P J Roy tatoc er B 979 4() 3 [8] VELEVA E Pliska tud Math Bulgar 8 (7) [9] VELEVA E Annuaire Univ oia Fac o Econ & Buss Admin 6 v [] VELEVA E Math and Education in Math 35(6) 3 36 Evelina Ilieva Veleva Deartment o Numerical Methods and tatistics Rouse University Rouse 77 tudentska 8 Bulgaria address: eveleva@abvbg 8