Chapter 10: Canonical Correlation Analysis

Abstract

We will keep utilizing the same notations in this chapter. More specifically, lower-case letters x, y, … will denote real scalar variables, whether mathematical or random. Capital letters X, Y, … will be used to denote real matrix-variate mathematical or random variables, whether square or rectangular matrices are involved. A tilde will be placed above letters such as $$\tilde {x},\tilde {y},\tilde {X},\tilde {Y}$$ x ~ , y ~ , X ~ , Y ~ to denote variables in the complex domain. Constant matrices will for instance be denoted by A, B, C. A tilde will not be used on constant matrices unless the point is to be stressed that the matrix is in the complex domain. The determinant of a square matrix A will be denoted by |A| or det(A) and, in the complex case, the absolute value or modulus of the determinant of A will be denoted as |det(A)|. When matrices are square, their order will be taken as p × p, unless specified otherwise. When A is a full rank matrix in the complex domain, then AA∗ is Hermitian positive definite where an asterisk designates the complex conjugate transpose of a matrix. Additionally, dX will indicate the wedge product of all the distinct differentials of the elements of the matrix X. Letting the p × q matrix X = (xij) where the xij’s are distinct real scalar variables, $$\mathrm {d}X=\wedge _{i=1}^p\wedge _{j=1}^q\mathrm {d}x_{ij}$$ d X = ∧ i = 1 p ∧ j = 1 q d x i j . For the complex matrix $$\tilde {X}=X_1+iX_2,\ i=\sqrt {(-1)}$$ X ~ = X 1 + i X 2 , i = ( − 1 ) , where X1 and X2 are real, $$\mathrm {d}\tilde {X}=\mathrm {d}X_1\wedge \mathrm {d}X_2$$ d X ~ = d X 1 ∧ d X 2 .