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1: 35.1 Special Notation
a , b

complex variables.

Ω

space of positive-definite real symmetric matrices.

X > T

X - T is positive definite. Similarly, T < X is equivalent.

2: 35.5 Bessel Functions of Matrix Argument
35.5.3 B ν ( T ) = Ω etr ( - ( T X + X - 1 ) ) | X | ν - 1 2 ( m + 1 ) d X , ν , T Ω .
35.5.5 0 < X < T A ν 1 ( S 1 X ) | X | ν 1 A ν 2 ( S 2 ( T - X ) ) | T - X | ν 2 d X = | T | ν 1 + ν 2 + 1 2 ( m + 1 ) A ν 1 + ν 2 + 1 2 ( m + 1 ) ( ( S 1 + S 2 ) T ) , ν j , ( ν j ) > - 1 , j = 1 , 2 ; S 1 , S 2 𝒮 ; T Ω .
3: 35.3 Multivariate Gamma and Beta Functions
35.3.2 Γ m ( s 1 , , s m ) = Ω etr ( - X ) | X | s m - 1 2 ( m + 1 ) j = 1 m - 1 | ( X ) j | s j - s j + 1 d X , s j , ( s j ) > 1 2 ( j - 1 ) , j = 1 , , m .
4: 19.31 Probability Distributions
More generally, let A ( = [ a r , s ] ) and B ( = [ b r , s ] ) be real positive-definite matrices with n rows and n columns, and let λ 1 , , λ n be the eigenvalues of A B - 1 . …
5: 21.1 Special Notation
g , h

positive integers.

Ω

g × g complex, symmetric matrix with Ω strictly positive definite, i.e., a Riemann matrix.

6: 35.2 Laplace Transform
35.2.1 g ( Z ) = Ω etr ( - Z X ) f ( X ) d X ,
7: 35.6 Confluent Hypergeometric Functions of Matrix Argument
35.6.2 Ψ ( a ; b ; T ) = 1 Γ m ( a ) Ω etr ( - T X ) | X | a - 1 2 ( m + 1 ) | I + X | b - a - 1 2 ( m + 1 ) d X , ( a ) > 1 2 ( m - 1 ) , T Ω .
35.6.8 Ω | T | c - 1 2 ( m + 1 ) Ψ ( a ; b ; T ) d T = Γ m ( c ) Γ m ( a - c ) Γ m ( c - b + 1 2 ( m + 1 ) ) Γ m ( a ) Γ m ( a - b + 1 2 ( m + 1 ) ) , ( a ) > ( c ) + 1 2 ( m - 1 ) > m - 1 , ( c - b ) > - 1 .
8: 35.4 Partitions and Zonal Polynomials
9: 1.5 Calculus of Two or More Variables
and the second-order term in (1.5.18) is positive definite (negative definite), that is, …
10: 35.8 Generalized Hypergeometric Functions of Matrix Argument