### 1.4.2 Direct Sums and Products The direct sum of two matrices A and B, denoted by A ⊕ B, is a block diagonal matrix formed by placing A and B on the diagonal. If A is an m × n matrix and B is a p × q matrix, then A ⊕ B is an (m + p) × (n + q) matrix. The direct product of two matrices A and B, denoted by A ⊗ B, is a block matrix formed by multiplying each element of A by the matrix B. If A is an m × n matrix and B is a p × q matrix, then A ⊗ B is an mp × nq matrix. ### 1.4.3 The Vec(·) and Vech(·) Operators The vec(·) operator is a matrix operator that stacks the columns of a matrix into a single column vector. For example, if A is an m × n matrix, then vec(A) is an mn × 1 column vector formed by stacking the columns of A. The vech(·) operator is a matrix operator that stacks the columns of a matrix into a single column vector, but only the lower triangular part of the matrix is included. For example, if A is an m × n matrix, then vech(A) is an (m(m+1))/2 × 1 column vector formed by stacking the columns of A, but only the lower triangular part of the matrix is included. ### 1.4.4 Matrix Differentiation The derivative of a scalar function with respect to a matrix is a matrix whose elements are the partial derivatives of the scalar function with respect to the elements of the matrix. The derivative of a vector function with respect to a matrix is a matrix whose elements are the partial derivatives of the vector function with respect to the elements of the matrix. The derivative of a matrix function with respect to a matrix is a matrix whose elements are the partial derivatives of the matrix function with respect to the elements of the matrix. ### 1.5 Distributions In this section, we review some basic results in multivariate distribution theory. We assume the reader has a basic working knowledge of multivariate distribution theory and so the results presented here are stated without proof. #### 1.5.1 General Results The multivariate normal distribution is a continuous probability distribution that is often used to model the distribution of a random vector. The multivariate normal distribution is characterized by its mean vector and covariance matrix. The multivariate normal distribution is a generalization of the univariate normal distribution to multiple dimensions. #### 1.5.2 The Multivariate Normal Distribution The multivariate normal distribution is a continuous probability distribution that is often used to model the distribution of a random vector. The multivariate normal distribution is characterized by its mean vector and covariance matrix. The multivariate normal distribution is a generalization of the### 1.4.2 Direct Sums and Products The direct sum of two matrices A and B, denoted by A ⊕ B, is a block diagonal matrix formed by placing A and B on the diagonal. If A is an m × n matrix and B is a p × q matrix, then A ⊕ B is an (m + p) × (n + q) matrix. The direct product of two matrices A and B, denoted by A ⊗ B, is a block matrix formed by multiplying each element of A by the matrix B. If A is an m × n matrix and B is a p × q matrix, then A ⊗ B is an mp × nq matrix. ### 1.4.3 The Vec(·) and Vech(·) Operators The vec(·) operator is a matrix operator that stacks the columns of a matrix into a single column vector. For example, if A is an m × n matrix, then vec(A) is an mn × 1 column vector formed by stacking the columns of A. The vech(·) operator is a matrix operator that stacks the columns of a matrix into a single column vector, but only the lower triangular part of the matrix is included. For example, if A is an m × n matrix, then vech(A) is an (m(m+1))/2 × 1 column vector formed by stacking the columns of A, but only the lower triangular part of the matrix is included. ### 1.4.4 Matrix Differentiation The derivative of a scalar function with respect to a matrix is a matrix whose elements are the partial derivatives of the scalar function with respect to the elements of the matrix. The derivative of a vector function with respect to a matrix is a matrix whose elements are the partial derivatives of the vector function with respect to the elements of the matrix. The derivative of a matrix function with respect to a matrix is a matrix whose elements are the partial derivatives of the matrix function with respect to the elements of the matrix. ### 1.5 Distributions In this section, we review some basic results in multivariate distribution theory. We assume the reader has a basic working knowledge of multivariate distribution theory and so the results presented here are stated without proof. #### 1.5.1 General Results The multivariate normal distribution is a continuous probability distribution that is often used to model the distribution of a random vector. The multivariate normal distribution is characterized by its mean vector and covariance matrix. The multivariate normal distribution is a generalization of the univariate normal distribution to multiple dimensions. #### 1.5.2 The Multivariate Normal Distribution The multivariate normal distribution is a continuous probability distribution that is often used to model the distribution of a random vector. The multivariate normal distribution is characterized by its mean vector and covariance matrix. The multivariate normal distribution is a generalization of the