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Numerical Methods for Computing Angles Between Linear Subspaces

Åke Björck and Gene H. Golub
Mathematics of Computation
Vol. 27, No. 123 (Jul., 1973), pp. 579-594
DOI: 10.2307/2005662
Stable URL: http://www.jstor.org/stable/2005662
Page Count: 16
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Abstract

Assume that two subspaces $F$ and $G$ of a unitary space are defined as the ranges (or null spaces) of given rectangular matrices $A$ and $B$. Accurate numerical methods are developed for computing the principal angles $\theta_k(F, G)$ and orthogonal sets of principal vectors $u_k \in F$ and $\upsilon_k \in G, k = 1, 2, \cdots, q = \dim(G) \leqq \dim(F)$. An important application in statistics is computing the canonical correlations $\sigma_k = \cos \theta_k$ between two sets of variates. A perturbation analysis shows that the condition number for $\theta_k$ essentially is $\max(\kappa(A), _\kappa(B))$, where $\kappa$ denotes the condition number of a matrix. The algorithms are based on a preliminary $QR$-factorization of $A$ and $B$ (or $A^H$ and $B^H$), for which either the method of Householder transformations (HT) or the modified Gram-Schmidt method (MGS) is used. Then $\cos \theta_k$ and $\sin \theta_k$ are computed as the singular values of certain related matrices. Experimental results are given, which indicates that MGS gives $\theta_k$ with equal precision and fewer arithmetic operations than HT. However, HT gives principal vectors, which are orthogonal to working accuracy, which is not generally true for MGS. Finally, the case when $A$ and/or $B$ are rank deficient is discussed.

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