"Frontmatter". In: Analysis of Financial Time Series

APPENDIX A. REVIEW OF VECTORS AND MATRIXES 351Therefore, 3 and 1 are eigenvalues of Σ with normalized eigenvectors ( √ 1 , √ 1 ) ′ and2 2( √ 1 , − √ 1 ) ′ , respectively. It is easy to verify that the spectral decomposition holds—2 2that is,⎡⎣1√21√2⎤√2 1⎦√−12[ 2 11 2] ⎡ ⎣√12√12⎤√2 1√−12⎦ =[ ] 3 0.0 1For a symmetric matrix A, there exists a lower triangular matrix L with diagonalelements being 1 and a diagonal matrix G such that A = LGL ′ ; see Chapter 1 ofStrang (1980). If A is positive definite, then the diagonal elements of G are positive.In this case,A = L √ G √ GL ′ = (L √ G )(L √ G ) ′ ,where L √ G is again a lower triangular matrix and the square root is taking elementby element. Such a decomposition is called the Cholesky Decomposition of A. Thisdecomposition shows that a positive definite matrix A can be diagonalized asL −1 A(L ′ ) −1 = L −1 A(L −1 ) ′ = G.Since L is a lower triangular matrix with unit diagonal elements, L −1 is also lowertriangular matrix with unit diagonal elements. Consider again the prior 2 × 2matrixΣ.ItiseasytoverifythatL =[ 1.0] 0.00.5 1.0and G =[ 2.0] 0.00.0 1.5satisfy that Σ = LGL ′ . In addition,L −1 =[ 1.0] 0.0−0.5 1.0and L −1 Σ(L −1 ) ′ = G.Vectorization and Kronecker ProductWriting a m × n matrix A in its columns as A =[a 1 ,...,a n ],wedefine the stackingoperation as vec(A) = (a ′ 1 , a′ 2 ,...,a′ m )′ , which is a mn × 1 vector. For two matrixesA m×n and C p×q , the Kronecker product between A and C is⎡⎤a 11 C a 12 C ··· a 1n Ca 21 C a 22 C ··· a 2n CA ⊗ C = ⎢⎥⎣ . .. ⎦a m1 C a m2 C ··· a mn Cmp×nq.