Normal Multivariate Analysis and the Orthogonal Group

By A. James. 1954

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On the largest principal angle between random subspaces (2006, methodology): “probability measure invariant under the transformations of Grass(p,n) induced by orthogonal transformations of Rn; see [ Jam54 , §4.6]).”
A statistical model for random rotations (2006, methodology): “Following James [6], an invariant measure on SO(p) is given by the exterior product of the elements of S−1(P ′dP) = S−1{2(Ip + S)−1dS(Ip − S)−1)}.”
The Tracy–Widom limit for the largest eigenvalues of singular complex Wishart matrices (2008, background): “Consider the following analytic homeomorphism g (see [11] for a useful summary of concepts from differential geometry) of almost all of the U(p) on almost all the product CV (p,n)× U(p − n).”
Automatic Choice of Dimensionality for PCA (2000, background): “The prior density for U is the reciprocal of the area of the manifold [7]:”
MIMO Transceiver Design via Majorization Theory (2007, background): “In other words, when Ω goes over the whole Stiefel manifold [68]”
BAYESIAN REFERENCE ANALYSIS OF COINTEGRATION (2005, background): “…in a 1-1 correspondence with ther-dimensional subspaces ofRp+ The set ofr-dimensional subspaces ofRp is an analytic manifold of dimension~ p 2 r !r ~James, 1954!, which has been termed theGrassman manifoldand is denoted byGr, p2r + The uniform distribution onGr, p2r is naturally defined as…”