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Bain, L. J. & Engelhardt, M. (1989), Introduction to Probability and Mathematical Statistics, PWS-KENT Publishing Company.

Brillinger, D. R. (1975), TIme Series, Data Analysis and Theory, Holt, Rinehart and Winston.

Conway, J. B. (1978), Functions of One Complex Variable, Graduate Texts in Mathematics 11, 2nd edn, Springer Verlag.

Cramer, H. (1945), Mathematical Methods of Statistics, Almqvist & Wiksells.

Darroch, J. N., Lauritzen, S. L. & Speed, T. P. (1980), 'Markov fields and log-linear interaction models for contingency tables', The Annals of Statistics 8,522-539.

Dawid, A P. (1979), 'Conditional independence in statistical theory (with discussion)', Journal of the Royal Statistical Society 41, 1-3\.

Dempster, A P. (1972), 'Covariance selection', Biometrics 28,157-175.

Eaton, M. L. (1983), Multivariate Statistics, A Vector Space Approach, John Wiley & Sons.

Edwards, C. H. & Penney, D. E. (1988), Elementary Linear Algebra, Prentice Hall International.

Eriksen, P. S. (1992), 'Covariance selection and graphical chain models, draft version', Depart-ment of Mathematics and Computer Science, Aalborg University. Lecture Notes.

Frederiksen, E. & Ragnarsson, R. (1986), En stokastisk model for mobil radio kommunikation, Master's thesis, Department of Mathematics and Computer Science, Aalborg University. In Danish.

Giri, N. (1965), 'On the complex analogues of t2_ and r 2-tests', The Annals of Mathematical Statistics 36, 664-670.

Goodman, N. R. (1963), 'Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction), , The Annals of Mathematical Statistics 34, 152 - 177.

Gupta, A K. (1971), 'Distribution of Wilks' likelihood-ratio criterion in the complex case', Annals of the Institute of Statistical Mathematics 23, 77-78.

Halmos, P. R. (1974), Finite-Dimensional Vector Spaces, Undergraduate Texts in Mathematics, Springer Verlag. Reprint of the 2nd edition published in 1958 by D. Van Nostrand Company, Princeton, N. J .. , in series: The University Series in Undergraduate Mathematics.

Jensen, J. L. (1991), 'A large deviation-type approximation for the "Box class" ofthe likelihood ratio criteria' , Journal of the American Statistical Association 86, 437-440.

Khatri, C. G. (1965a), 'Classical statistical analysis based on a certain multivariate complex distribution', The Annals of Mathematical Statistics 36, 98-114.

Khatri, C. G. (1965b), 'A test for reality of a covariance matrix in a certain complex Gaussian distribution', The Annals of Mathematical Statistics 36, 115-119.

Krishnaiah, P. R. (1976), 'Some recent developments on complex multivariate distributions', Journal of Multivariate Analysis 6, 1-30.

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Lauritzen, S. L. (1989), Lectures on Contingency Tables, 3rd edn, Department of Mathematics and Computer Science, Aalborg University.

Lauritzen, S. L. (1993), 'Graphical association models, draft version', Department of Mathe­matics and Computer Science, Aalborg University.

Lauritzen, S. L. & Frydenberg, M. (1989), 'Decomposition of maximum likelihood in mixed graphical interaction models', Biometrika 76, 539-555.

Leimer, H.-G. (1989), 'Triangulated graphs with marked vertices', Annals of Discrete Mathe-matics 41, 311-324.

MacDuffee, C. C. (1956), The Theory of Matrices, Chelsea Publishing Company, New York.

Muirhead, R. J. (1982), Aspects of Multivariate Statistical Theory, John Wiley & Sons.

Pearl, J. (1988), Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufmann, San Mateo.

Pearl, J. & Paz, A. (1986), Graphoids: A graph based logic for reasoning about relevancy relations, in 'Proceedings of the European Conference on Artificial Intellligence, Brighton, United Kingdom'.

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Wermuth, N. (1976), 'Analogies between multiplicative models in contingency tables and covariance selection', Biometrics 32, 95-108.

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A

Complex Matrices

This appendix contains results on complex matrices. It is rather elementary and it is only meant as help for readers who are not familiar with complex matrix algebra. The proofs are omitted but can be found in Eaton (1983), Halmos (1974), MacDuffee (1956), Giri (1965) or Goodman (1963).

A.1 Complex Vector Space

Definition A.I Vector Space Let F be a field. A set V is called a vector space over F if the following axioms hold.

1. For all elements c, d E V there corresponds an element c + dE V, called the sum, such that

i. addition is commutative, c + d = d + c.

ii. addition is associative, (c + d) + f = c + (d + f) for f E V.

iii. there exists a unique element 0 E V such that c + 0 = cfor all c E V. The element o is called the zero element.

iv. for all c E V there corresponds a unique element -c such that c + (-c) = o. The element -c is called the additive inverse.

2. For all a E F and all c E V there corresponds an element ac E V, called the product of a and c, such that

i. multiplication by scalars is associative, a (f3c) = (af3) cfor f3 E F.

ii. there exists a unique element I E F, such that Ic = cfor all c E V. The element 1 is called the multiplicative identity.

iii. multiplication by scalars is distributive w.r.t. addition of elements in F, (a + {3) c = ac + f3cfor f3 E F.

iv. multiplication by scalars is distributive w.r.t. addition of elements in V, a (c + d) = ac+ adford E V.

If F in Definition A.l is the field of real numbers, lR, the set V is called a real vector space and if F is the field of complex numbers, C, we call V a complex vector space.

165

166 Appendix A. Complex Matrices

Definition A.2 Complex vector A vector c = (Ck). where Ck E IC for k = 1,2, ... , p, is called a complex vector.

The set of p-dimensional complex vectors, denoted by ICP, is a p-dimensional complex vector space since the axioms for a vector space over IC are fulfilled. Note that ICP also can be considered as a real vector space. The standard basis for ICP considered as a complex vector space is (el, e2,' .. , ep ), where ej has the j'th element equal to one and the remaining elements equal to zero.

A.2 Basic Operations of Complex Matrices

Definition A.3 Complex matrix An n x p array C = (Cjk), where Cjk E IC for j = 1,2, ... , nand k = 1,2, ... ,p, is called an n x p complex matrix.

The set of all n x p complex matrices is denoted by IC nxp.

Definition A.4 Matrix addition Let C = (Cjk) and D = (djk ) be n x p complex matrices, then

is the sum ofC and D, where C + DE ICnxp.

Definition A.S Matrix multiplication Let C = (Cjk) E ICnxp and DE (dkd E ICpxm, then

CD = ((t Cjkdkt) ) k=l jt

is the product ofC and D, where CD E ICnxm.

Notice that DC is not defined.

Theorem A.1 Rules for complex matrix algebra For suitable complex matrices and Ct, f3 E IC the following properties hold.

1. C+D=D+C.

2. (C+D)+F=C+(D+F).

3. C (D + F) = CD + CF.

4. (C+D)F = CF+DF.

5. C(DF) = (CD)F.

6. a ({jC) = (a{j) C = ({ja) C = {j (aC).

7. (a + (j) C = aC + {jC.

B. a(C+D)=aC+aD.

9. a (CD) = C (aD) = (aC) D.

A.3. Inverse Matrix 167

With the definition of matrix addition and some of the rules of Theorem A.I we see that the elements of C nxp fulfill the axioms for a vector space over C. Hence C nxp is a complex vector space. Note that C nxp also can be considered as a real vector space.

A.3 Inverse Matrix

Definition A.6 Inverse Matrix Let C E CpxP. If there exists aD E Cpxp such that CD = DC = 1, then D is called the inverse ofC and is denoted by c-I.

Theorem A.2 Rules for inverse matrix For suitable C, DE Cpxp the following properties hold.

1. (C-Ifl = C.

2. (CD)-I = D-IC-I.

A.4 Determinant and Eigenvalues

Let (iI,j2,'" ,jp) be a permutation of the first p positive integers. Let ak be the number of integers following jk that are smaller than jk' Note that a p is always zero. The sum Et=1 ak is called the number of inversions in the permutation (jl, j2, . .. , jp). A permutation is said to have even or odd parity according to whether its number of inversions is even or odd. We define

6 (.. . ) = { 1 if the parity of (ji> j2, ... , jp) is even JI,J2, .. · ,]p -1 if the parity of (ji>h, ... ,jp) isodd

which enables us to define the determinant of a square complex matrix.

168 Appendix A. Complex Matrices

Definition A.7 Determinant The transformation det : CPxp r-+ C is called the determinant andfor a p x p complex matrix C = (Cjk) it is given by

det (C) = L <> (jl, h, ... ,jp) Clj! C2ja ... Cpjp ,

where the sum is over all possible permutations (jl, j2, . .. ,jp) of {I, 2, ... ,p}.

Definition A.8 Eigenvalue Let C E CpxP. A number A E C is called an eigenvalue ofC if

(A.i) det (Up - C) = 0 .

Equation (A. 1 ) is called the characteristic equation of C. It is a p'th degree polynomial in A. Hence a p x p complex matrix has p eigenvalues.

Theorem A.3 Rules of determinant For C, DE Cpxp the following properties hold.

1. det (CD) = det (C) det (D).

2. det (C) i= 0 iff the columns of C are linear independent vectors in Cp.

3. If AI, A2, . .. ,Ap are the eigenvalues ofC, then det (C) = n~=l Aj.

4. If AI, A2, .. . ,Ap are the eigenvalues ofC, then det (Ip + C) = m=l (1 + Aj).

5. The eigenvalues of CD are the same as the eigenvalues of DC.

Definition A.9 Nonsingular Let C E Cpxp. If det (C) i= 0, then C is nonsingular.

Notice that C is nonsingular iff C- l exists.

A.S Trace and Rank

Definition A.tO Trace operator Let C = (Cjk) E CpxP. The trace operator, tr : Cpxp r-+ C, ofC is given by

p

tr (C) = L Cjj . j=l

A. 6. Conjugate Transpose Matrix 169

The value of tr (C) is called the trace of C.

Theorem A.4 Rules for trace For C, DE Cpxp the following properties hold.

1. tr (C + D) = tr(C) +tr(D).

2. tr (CD) = tr (DC).

3. If Al, A2, ... ,Ap are the eigenvalues ofC. then tr (C) = E~=l Aj.

Definition A.ll Rank The maximum number of linear independent columns of a complex matrix C is called the rank of the matrix and is denoted by rank (C).

A.6 Conjugate Transpose Matrix

Definition A.12 Conjugate transpose matrix The conjugate transpose of an n x p complex matrix C = (Cjk) is the p x n complex matrix given by C" = (ckj)fork = 1,2, ... ,pandj = 1,2, ... ,no

Rules for the conjugate transpose operation are equivalent to the rules for the transpose operation for real matrices.

Theorem A.S Rules for the conjugate transpose operation For suitable complex matrices and C E C the following properties hold.

1. (C"r = C.

2. (cCr = cC".

3. (C+Dr=C"+D".

4. (CDr = D"C".

5. det (C") = det (C).

6. det({C}) = det (C) det (C").

7. tr (C") = tr (C).

8. (C-lf = (C")-l.

9. rank (C) = rank (C") = rank (CC") = rank (C"C).

170 Appendix A. Complex Matrices

A.7 Hermitian Matrix

An important property of a complex matrix is that it can be Hermitian. A complex matrix being Hermitian corresponds to a real matrix being symmetric.

Definition A.13 Hermitian Let C E Cpxp. IfC = C', then C is Hermitian.

The set of all p x p Hermitian matrices is denoted by Cjpp. Notice that this set satisfies the axioms of a vector space over C, hence C jpP is a complex vector space.

Definition A.14 Skew Hermitian Let C E Cpxp. /fC = -C', then C is skew Hermitian.

Theorem A.6 Rules for Hermitian matrices For C E Cpxp the following properties hold.

1. Let C = A + iB, where A, B E ]RPXp. If C is Hermitian, then A is symmetric and B is skew symmetric.

2. C is Hermitian iff {C} is symmetric.

3. If C is Hermitian, then det (C)2 = det ({ C}).

4. The eigenvalues of a Hermitian matrix are real.

5. The eigenvalues of a skew Hermitian matrix are imaginary.

6. Let C be Hermitian. The eigenvalues ofC are all equal to zero iffC = O.

7. Let C be Hermitian. The eigenvalues ofC are all equal to one iffC = IV'

8. IfC is Hermitian and rank (C) = 1, then det (Ip + C) = 1 + tr (C).

A.8 Unitary Matrix

Definition A.IS Unitary Let C E Cpxp be nonsingular. /fC' = C-\ then C is unitary.

Notice that a complex matrix being unitary corresponds to a real matrix being orthogonal.

A. 9. Positive Semidefinite Complex Matrices 171

Theorem A.7 Rules for unitary matrices For C E Cpxp the/ollowing properties hold.

1. C is unitary iff {C} is orthogonal.

2. If C is Hermitian, then it holds that there exists a unitary p x p matrix U such that U·CU = diag (A1> A2,'" ,Ap), where Aj/or j = 1,2, ... ,p are the eigenvalues o/C.

A.9 Positive Semidefinite Complex Matrices

Definition A.16 Positive semidefinite complex matrix Let C E C);p. If c·Cc ~ O/or all c E CP, then C is positive semidefinite. This is denoted by C~O.

The set of all p x p positive semidefinite complex matrices is denoted by C~xp.

Theorem A.S Rules for the positive semidefinite complex matrices For C E C);P the/ollowing properties hold.

1. C ~ 0 with rank (C) = r iff there exists aD E Cpxp with rank (D) = r such that C=D·D.

2. IfC ~ 0 and DE cnxp , then DCD· ~ O.

3. C ~ 0 iff {C} ~ O.

4. C ~ 0 iff the eigenvalues o/C are nonnegative.

5. IfC ~ 0, then tr (C) ~ O.

A.10 Positive Definite Complex Matrices

Definition A.17 Positive definite complex matrix Let C E C);p. If c·Cc > O/or all c E CP \ {O}, then C is positive definite. This is denoted byC > O.

The set of all p x p positive definite complex matrices is denoted by C~p.

172 Appendix A. Complex Matrices

Theorem A.9 Rules for the positive definite complex matrices For C E C ~p the following properties hold.

1. C?:: 0 and C is nonsingular iff C > O.

2. C> 0 iffC-1 > O.

3. C > 0 iff there exists a nonsingular complex matrix D E Cpxp such that C = DD·.

4. /f C > 0, then there exists a complex matrix ct > 0 such that C = (ct) 2.

5. /fC> 0 and DE C nxp withfull rank, then DCD' > O.

6. C > 0 iff {C} > O.

7. C > 0 iff the eigenvalues of C are positive.

8. /fC > 0, then tr (C) > O.

9. 1fC> 0 and aC + bD > 0 for all a, bE 114 and DE CpxP , then D ?:: O.

10. /fC> 0, D E C~xP and a E lR+, then C + aD> O.

A.11 Direct Product

Definition A.tS The direct product of matrices Let C = (Cjk) and D = (drs) be n x p and m x q complex matrices, respectively. The direct product C ® D is the nm x pq matrix with elements given as

Theorem A.tO Rules for the direct product For suitable complex matrices the direct product satisfies the following properties.

1.0®C=C®0=0.

2. C® (D +E) = C®D+C®E.

3. (C + D) ® E = C ® E + D ® E.

4. (cC) ® (dD) = (cd) (C ® D), where c, dEC.

5. (C ® Dr = C· ® D'.

6. (C ® D) (E ® F) = (CE) ® (DF).

7. If the inverse complex matrices C- l and D-1 exist, then (C ® D)-l = C- l ® D-1.

A. 12. Partitioned Complex Matrices 173

8. If C and D are square matrices, then tr (C ® D) = tr ( C) tr(.D).

9. IfC E c nxn and DE cP xP, then det (C ® D) = det (C)P det (Dr.

10. IfC and D both are positive (semi)definite matrices, then so is C ® D.

11. If C and D are orthogonal projections, then so is C ® D.

A.12 Partitioned Complex Matrices

Theorem A.ll Let C E C pxp be partitioned as

where C jk has dimension Pj x Pk for j, k = 1,2, and PI + P2 = p. Further let C U .2

C u - CI2C221C21 andC22.1 = C 22 - C2IClIICI2'

1. If all the inverses exist, then

or

2. Ifp = 2, then

C-I ___ 1_ ( C 22

- det (C) -C21

3. IfCu is nonsingular, then

det (C) det (C22.1 ) det (Cu ) .

4. If C 22 is nonsingular, then

det (C) = det (CU '2 ) det (C22 )

5. If C u is nonsingular and C 21 = Ci2' then

C 2 0 iffCu 20 andC22.1 20.

174 Appendix A. Complex Matrices

6. If C 22 is nonsingular and C 21 = C;:2' then

C 2 0 iffC22 2 OandC l1 .2 20.

C> 0 iffC l1 > 0 and C 22.1 > 0 .

C> 0 iffC22 > OandCl1 .2 > O.

9. IfC 2 0, then C l1 20 and C 22 2 o.

10. IfC> 0, then C l1 > 0 and C 22 > O.

B

Orthogonal Projections

This appendix contains results on orthogonal projections. The proofs are omitted but can be found in Eaton (1983) and MacDuffee (1956).

Definition B.I Orthogonal projection Let Nand N 1. be subspaces in C n such that N E9 N 1. = C n and N 1. is the orthogonal complement of N w.r.t. the inner product on C n. If Z = Y + z E C n with yEN and zEN 1..

then y is called the orthogonal projection of Z onto N and z is called the orthogonal projection ofz onto N1..

TbeoremB.I For P N E c nxn thefollowing properties hold.

1. Let z = y + z E Cn with yEN and z E N1.. If PN is a complex matrix such that y = PNz. then P N represents the orthogonal projection ofcn onto N. The complex matrix P N is also called a projection matrix.

2. Let P N represents the orthogonal projection of C n onto N. Then In - P N represents the orthogonal projection of en onto N1.. The complex matrix In - P N is also denoted byP~.

3. If P N is an idempotent Hermitian matrix. i.e. P N = P~ = Piv, then P N represents the orthogonal projection ofcn onto R[PN].

4. If P N is a projection matrix. then rank (PN) = tr (PN).

5. Let P N represent the orthogonal projection ofcn onto N. Then R[PN] = N and the dimension of N is tr (PN).

6. Let P N E c nxn be Hermitian. The complex matrix P N is idempotent of rank r iff it has r eigenvalues equal to one and n - r equal to zero.

7. IfZ E C nxp• R[Z] = Nand Z hasfull rankp. then a matrix representing the orthogonal projection ofcn onto N is given by P N = Z (Z" Z)-l Z".

8. If Z E C nxp , R [Z ® Ip] = M and Z has full rank p. then a matrix representing the orthogonal projection ofC nxp onto M is given by PM = Z (Z" Z)-l Z" ® Ip.

9. The vector space No is a subspace of N iff P N P No = P NoP N = P No·

175

Index

C-marginal adjusting operator, 130 Q-regularity, 129

A absolute value, 1 adjacency, 85

B beta distribution, 56, 65

product, 57 boundary,86

C characteristic function, 7, 12, 20, 23, 33 chi-square distribution, 40

product, 51 chord,88 clique, 91 closure, 86 collapsibility, 94 complete graph, 87 complete subset, 87 complex U -distribution

definition, 55 independence, 57,64

complex covariance structure, 11 complex MANOVA models

definition, 67 distribution of maximum likelihood estima-

tors, 73 maximum likelihood estimators, 70, 73 test concerning the mean, 75, 78 test for independence, 80

complex matrix, 2, 166 complex normal decomposable models, 145 complex normal distribution

multivariate, 22 univariate, 15

complex normal graphical models C-marginal adjusting operator, 130 decomposable, 145 decomposition, 137, 142 definition, 118 hypothesis testing, 147

177

IPS-algorithm, 130 likelihood equations, 128 maximum likelihood estimation, 121

complex random matrix covariance,9 definition, 8 expectation, 8 variance, 9

complex random variable covariance, 5 definition, 5 expectation, 5 variance, 6

complex random vector conditional independence, 100, 103 covariance,9 definition, 8 expectation, 8 mutual independence, 13 variance, 9

complex vector, I, 166 complex vector space, I, 165 complex Wishart distribution

definition, 40 density function, 47 independence,43,44 mean,40 partitioning, 44 positive definite, 44 sum, 43

concentration matrix, 116 conditional density function, 99 conditional distribution, 31, 37

definition, 100 of a transformation, 100

conditional independence concentration matrix, 117 definition, 100 factorization criterion, 102, 104 properties, 101-106

conditional independence graph, 108 conjugate transpose, 169 connectivity components, 89

178 Index

correspondence between complex Wishart and chi-square distribution, 40

correspondence between complex U -distribution and beta distribution, 57, 65

correspondence between complex and real nor­mal distribution, 16, 19,25

covariance, 5, 9 rules,6,10

cycle, 88

D decomposability, 91, 93, 95 decomposition, 90 density function, 18,20,26,33,47

conditional, 99 design matrix, 68 detenninant, 168 direct product, 172

definition, 2 interpretation, 2

direct union, 87, 96

E edge, 85 eigenvalue, 168 expectation, 5, 8

rules,6,10

F factorization property, 106

decomposition, 112

G global Markov property, 107 graph,85

adjacent, 85 boundary, 86 chord,88 clique, 91 closure, 86 collapsibility, 94 complete, 87 conditional independence, 108 connectivity components, 89 cycle, 88 decomposability, 91, 93, 95 decomposition, 90 direct union, 87, 96 edge, 85 induced subgraph, 86

H

intersection, 87 maximum cardinality search, 92 MCS-algorithm, 92 neighbours, 85 path,88 regular edge, 94, 95 RIP-ordering, 92 running intersection property ordering, 92 separation, 89 union, 87 vertex, 85

Hennitian, 170 hypothesis testing

complex MANOVA models, 75, 78 decomposable models, 159 graphical models, 147 regular edge, 152

I independence

in the complex U -distribution, 57, 64 in the complex normal distribution, 28-30,

34--36 in the complex Wishart distribution, 43, 44 mutual,13 of maximum likelihood estimators, 73

induced subgraph, 86 inner product, I, 2 intersection, 87 inverse complex matrix, 167 IPS-algorithm, 129, 130 isomorphism

vector space, 3 iterative proportional scaling, 129, 130

K Kronecker product, 2

L law of total probability, 100 likelihood equations, 128 local Markov property, 107

M marginal distribution, 29, 36 Markov property, 109, 110

global,107 local,I07

pairwise, 107 matrix addition, 166 matrix multiplication, 166 maximum cardinality search, 92 maximum likelihood estimation

complex MANOVA models, 70 concentration matrix, 128, 131 decomposition, 142 graphical models, 121 likelihood equations, 128, 131

maximum likelihood estimators, 76, 73 distribution of, 73 independence, 73 unbiased, 74

MCS-algorithm, 92 mean, 5, 8

rules, 6, 10 multiple time series

example, 53, 54, 83, 120 multivariate complex normal distribution

characteristic function, 23, 33 conditional distribution, 31, 37 definition, 22, 32 density function, 26, 33 independence, 28-30, 34-36 marginal distribution, 29, 36 matrix notation, 32, 34 properties, 23, 34 reproductivity property, 27 special variance structure, 34

multivariate linear complex normal models, 67 complex MANOVA models, 67

N neighbours, 85 nonsingular, 168 normal equations, 73

o orthogonal projections, 175

p pairwise Markov property, 107 parameter set, 68 partitioned complex matrix

determinant, 173 in verse, 1 73 positive definite, 174 positive semidefinite, 174

path, 88

Index 179

positive ,'efinite, 44,53,171,174 positive semidefinite, 11, 171, 174

Q quadratic form, 41, 53, 64

R rank, 169 real vector space, 165 regular edge, 94, 95 reproductivity property, 21, 27 RIP-ordering, 92 rotation invariance, 17 running intersection property ordering, 92

S separation, 89 simple undirected graph, 85 skew Hermitian, 170

T tensor product, 2 trace, 168

U union, 87 unitary, 170 univariate complex normal distribution

V

arbitrary, 18 characteristic function, 20 definition, 19 density function, 20 reproductivity property, 21

standard, 15 definition, 16 density function, 18 rotation invariance, 17

variance, 6, 9 rules, 7, 11

vector space, 165 vector space isomorphism, 3 vertex, 85

Notation

The list given below contains symbols used in the book. For each symbol a short explanation is stated and if necessary pagereferences for further information are given in brackets.

Sets

lR R+ JRI' lRnxp

C CP cnxp

C~P C~xp

C~xp

CH(Q)

Cs(Q)

field of real numbers. set of nonnegative real numbers. real vector space of p-dimensional real vectors. real vector space of n x p real matrices. field of complex numbers. complex vector space of p-dimensional complex vectors. complex vector space of n x p complex matrices. complex vector space of p x p Hermitian matrices. set of p x p positive semidefinite complex matrices. set of p x p positive definite complex matrices. set of IVI x IVI Hermitian matrices, which contains zero entries according to missing edges in 9 = (V, E), (125). set of IVI x IVI positive semidefinite complex matrices, which contains zero entries according to missing edges in 9 = (V, E), (129). set of IVI x IVI positive definite complex matrices, which contains zero entries according to missing edges in 9 = (V, E), (118). vector space over C of complex random "variables" with finite second moment, (5,8).

Complex Numbers

c Re(·) Im(·) 1·1

imaginary unit. complex conjugate of c. real part. imaginary part. absolute value.

181

182 Notation

Matrix Algebra

(', .) {-}

[.J H

". " R[·J tr(· ) det(·) rank(·) AT C' C- I

C®D In In o diag(· .. ) C?O C>O P N

P~

Distributions

lEO V(.) q.,,) .cO N(fJ, ()"2)

N p (8, ~)

CN(fJ, ()"2)

CNp (8,H)

inner product, (1). 2p x 2p real matrix derived from a p x p complex matrix, (4). real vector space isomorphism between CP and ]R2P, (3). matrix obtained from a submatrix by filling in missing entries with zero entries, (116). length. range. trace. determinant. rank. transpose of a real matrix A. conjugate transpose of a complex matrix C. inverse of C. direct product of two complex matrices, (2). n x n identity matrix. n-dimensional vector of ones. matrix of zeros. diagonal matrix. C is positive semidefinite. C is positive definite. a matrix representing the orthogonal projection onto N. a matrix representing the orthogonal projection onto the orthogonal complement of N w.r.t. inner product, N 1..

expectation operator, (5, 8). variance operator, (6,9). covariance operator, (5,9) . distributional law. univariate real normal distribution with mean fJ and variance ()"2.

p-variate real normal distribution with mean 8 and variance matrix~. univariate complex normal distribution with mean fJ and variance ()"2.

p-variate complex normal distribution with mean 8 and variance matrix H.

CNnxp (8, J ® H) (n x p)-variate complex nonnal distribution with mean 8 and variance matrix J ® H.

CWp(H,n)

CU(p,m,n)

X~ B(n,p) Fn,p p

t.px

Ix exp(·) Xl Jl X 2

X I IX 2

complex Wishart distribution with dimension p, n degrees of freedom and mean nH. complex U-distribution with parameters p, m and n. chi-square distribution with k degrees of freedom. beta distribution with parameters n and p. F -distribution with parameters nand p. distribution. characteristic function of X. density function of X w.r.t. Lebesgue measure. exponential function. X I and X 2 are independent. Xl given X 2 •

Simple Undirected Graphs

Notation 183

9 = (V, E) gA

simple undirected graph with vertex set V and edge set E. subgraph induced by A.

f a "'p f3 a fp f3 bd(·) dO c c u n U C RIP MCS

1·1 Tc T

adjacent or neighbours. nonadjacent. there exists a path between a and f3. there exists no path between a and f3. boundary. closure. proper subset of. subset o.f union. intersection. direct union. set of cliques in a simple undirected graph. running intersection property, (92). maximum cardinality search, (92). cardinality. C -marginal adjusting operator, (130). adjusting operator for all the cliques, (131).

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