Linear and Graphical Models

1 Prerequisites.- 1.1 Complex Matrix Algebra.- 1.2 A Vector Space Isomorphism.- 1.3 Complex Random Variables.- 1.4 Complex Random Vectors and Matrices.- 2 The Multivariate Complex Normal Distribution.- 2.1 The Univariate Complex Normal Distribution.- 2.1.1 The Standard Complex Normal Distribution.- 2.1.2 The Complex Normal Distribution.- 2.2 The Multivariate Complex Normal Distribution.- 2.3 Independence, Marginal and Conditional Distributions.- 2.4 The Multivariate Complex Normal Distribution in Matrix Notation.- 3 The Complex Wishart Distribution and the Complex U-distribution.- 3.1 The Complex Wishart Distribution.- 3.2 The Complex U-distribution.- 4 Multivariate Linear Complex Normal Models.- 4.1 Complex MANOVA Models.- 4.2 Maximum Likelihood Estimation in Complex MANOVA Models.- 4.2.1 Distributions of the Maximum Likelihood Estimators.- 4.3 Hypothesis Testing in Complex MANOVA Models.- 4.3.1 Likelihood Ratio Test Concerning the Mean.- 4.3.2 Likelihood Ratio Test for Independence.- 5 Simple Undirected Graphs.- 6 Conditional Independence and Markov Properties.- 6.1 Conditional Independence.- 6.2 Markov Properties in Relation to Simple Undirected Graphs.- 7 Complex Normal Graphical Models.- 7.1 Notation.- 7.2 The Concentration Matrix.- 7.3 Complex Normal Graphical Models.- 7.4 Maximum Likelihood Estimation of the Concentration Matrix.- 7.4.1 Iterative Proportional Scaling.- 7.5 Decomposition of the Estimation Problem.- 7.5.1 Estimation in Complex Normal Decomposable Models.- 7.6 Hypothesis Testing in Complex Normal Graphical Models.- 7.6.1 Hypothesis Testing in Complex Normal Decomposable Models.- A Complex Matrices.- A.1 Complex Vector Space.- A.2 Basic Operations of Complex Matrices.- A.3 Inverse Matrix.- A.4 Determinant and Eigenvalues.- A.5 Trace and Rank.- A.6 Conjugate Transpose Matrix.- A.7 Hermitian Matrix.- A.8 Unitary Matrix.- A.9 Positive Semidefinite Complex Matrices.- A.10 Positive Definite Complex Matrices.- A.11 Direct Product.- A.12 Partitioned Complex Matrices.- B Orthogonal Projections.- Notation.