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35 Functions of Matrix ArgumentNotation

§35.1 Special Notation

(For other notation see Notation for the Special Functions.)

All matrices are of order m×m, unless specified otherwise. All fractional or complex powers are principal values.

a,b complex variables.
j,k,p,q nonnegative integers.
m positive integer.
[a]κ partitional shifted factorial (§35.4(i)).
𝟎 zero matrix.
𝐈 identity matrix.
𝓢 space of all real symmetric matrices.
𝐒,𝐓,𝐗 real symmetric matrices.
tr𝐗 trace of 𝐗.
etr(𝐗) exp(tr𝐗).
|𝐗| determinant of 𝐗 (except when m=1 where it means either determinant or absolute value, depending on the context).
|(𝐗)j| jth principal minor of 𝐗.
xj,k (j,k)th element of 𝐗.
d𝐗 1jkmdxj,k.
𝛀 space of positive-definite real symmetric matrices.
t1,,tm eigenvalues of 𝐓.
𝐓 spectral norm of 𝐓.
𝐗>𝐓 𝐗𝐓 is positive definite. Similarly, 𝐓<𝐗 is equivalent.
𝐙 complex symmetric matrix.
f(𝐗) complex-valued function with 𝐗𝛀.
𝐎(m) space of orthogonal matrices.
𝐇 orthogonal matrix.
d𝐇 normalized Haar measure on 𝐎(m).
Zκ(𝐓) zonal polynomials.

The main functions treated in this chapter are the multivariate gamma and beta functions, respectively Γm(a) and Bm(a,b), and the special functions of matrix argument: Bessel (of the first kind) Aν(𝐓) and (of the second kind) Bν(𝐓); confluent hypergeometric (of the first kind) F11(a;b;𝐓) or F11(ab;𝐓) and (of the second kind) Ψ(a;b;𝐓); Gaussian hypergeometric F12(a1,a2;b;𝐓) or F12(a1,a2b;𝐓); generalized hypergeometric Fqp(a1,,ap;b1,,bq;𝐓) or Fqp(a1,,apb1,,bq;𝐓).

An alternative notation for the multivariate gamma function is Πm(a)=Γm(a+12(m+1)) (Herz (1955, p. 480)). Related notations for the Bessel functions are 𝒥ν+12(m+1)(𝐓)=Aν(𝐓)/Aν(𝟎) (Faraut and Korányi (1994, pp. 320–329)), Km(0,,0,ν|𝐒,𝐓)=|𝐓|νBν(𝐒𝐓) (Terras (1988, pp. 49–64)), and 𝒦ν(𝐓)=|𝐓|νBν(𝐒𝐓) (Faraut and Korányi (1994, pp. 357–358)).