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
.