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11: 3.2 Linear Algebra
§3.2(iv) Eigenvalues and Eigenvectors
§3.2(vi) Lanczos Tridiagonalization of a Symmetric Matrix
The tridiagonal matrix
12: 1.2 Elementary Algebra
a real symmetric matrix if …an Hermitian matrix if …a tridiagonal matrix if … The matrix exponential is defined via …
13: 21.5 Modular Transformations
21.5.1 𝚪 = [ 𝐀 𝐁 𝐂 𝐃 ]
is a symplectic matrix, that is,
21.5.2 𝚪 𝐉 2 g 𝚪 T = 𝐉 2 g .
21.5.3 det 𝚪 = 1 ,
For a g × g matrix 𝐀 we define diag 𝐀 , as a column vector with the diagonal entries as elements. …
14: 21.10 Methods of Computation
§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface
In addition to evaluating the Fourier series, the main problem here is to compute a Riemann matrix originating from a Riemann surface. …
15: Donald St. P. Richards
Richards has published numerous papers on special functions of matrix argument, harmonic analysis, multivariate statistical analysis, probability inequalities, and applied probability. …
  • 16: 21.1 Special Notation
    g , h positive integers.
    𝛀 g × g complex, symmetric matrix with 𝛀 strictly positive definite, i.e., a Riemann matrix.
    A j k ( j , k ) th element of matrix 𝐀 .
    𝟎 g g × g zero matrix.
    𝐈 g g × g identity matrix.
    17: 35.12 Software
  • Koev and Edelman (2006). Computation of hypergeometric functions of matrix argument in MATLAB.

  • 18: 35.4 Partitions and Zonal Polynomials
    19: 32.4 Isomonodromy Problems
    32.4.4 𝐀 ( z , λ ) = ( 4 λ 4 + 2 w 2 + z ) [ 1 0 0 1 ] i ( 4 λ 2 w + 2 w 2 + z ) [ 0 i i 0 ] ( 2 λ w + 1 2 λ ) [ 0 1 1 0 ] ,
    32.4.5 𝐁 ( z , λ ) = ( λ + w λ ) [ 1 0 0 1 ] i w λ [ 0 i i 0 ] .
    32.4.6 𝐀 ( z , λ ) = i ( 4 λ 2 + 2 w 2 + z ) [ 1 0 0 1 ] 2 w [ 0 i i 0 ] + ( 4 λ w α λ ) [ 0 1 1 0 ] ,
    32.4.7 𝐁 ( z , λ ) = [ i λ w w i λ ] .
    32.4.8 𝐀 ( z , λ ) = [ 1 4 z 0 0 1 4 z ] + [ 1 2 θ u 0 u 1 1 2 θ ] 1 λ + [ v 0 1 4 z v 1 v 0 ( v 0 1 2 z ) / v 1 1 4 z v 0 ] 1 λ 2 ,
    20: 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 .
    35.3.3 B m ( a , b ) = 𝟎 < 𝐗 < 𝐈 | 𝐗 | a 1 2 ( m + 1 ) | 𝐈 𝐗 | b 1 2 ( m + 1 ) d 𝐗 , ( a ) , ( b ) > 1 2 ( m 1 ) .