Computes the Gram matrix of tangent vectors in the Manopt framework. function G = grammatrix(M, x, vectors) M is a Manopt manifold structure obtained from a factory. x is a point on the manifold M. vectors is a cell containing n tangent vectors at x. G is an n-by-n symmetric positive semidefinite matrix such that G(i, j) is the inner product between vectors{i} and vectors{j}, with respect to the metric on the tangent space to M at x. See also: orthogonalize tangentorthobasis

- orthogonalize Orthonormalizes a basis of tangent vectors in the Manopt framework.
- smallestinconvexhull Computes a minimal norm convex combination of given tangent vectors in Manopt.

0001 function G = grammatrix(M, x, vectors) 0002 % Computes the Gram matrix of tangent vectors in the Manopt framework. 0003 % 0004 % function G = grammatrix(M, x, vectors) 0005 % 0006 % M is a Manopt manifold structure obtained from a factory. 0007 % x is a point on the manifold M. 0008 % vectors is a cell containing n tangent vectors at x. 0009 % 0010 % G is an n-by-n symmetric positive semidefinite matrix such that G(i, j) 0011 % is the inner product between vectors{i} and vectors{j}, with respect to 0012 % the metric on the tangent space to M at x. 0013 % 0014 % See also: orthogonalize tangentorthobasis 0015 0016 % This file is part of Manopt: www.manopt.org. 0017 % Original author: Nicolas Boumal, April 28, 2016. 0018 % Contributors: 0019 % Change log: 0020 0021 0022 n = numel(vectors); 0023 0024 G = zeros(n); 0025 0026 for i = 1 : n 0027 0028 vi = vectors{i}; 0029 0030 G(i, i) = M.inner(x, vi, vi); 0031 0032 for j = (i+1) : n 0033 0034 vj = vectors{j}; 0035 G(i, j) = M.inner(x, vi, vj); 0036 0037 % Manopt is designed to work with real inner products, 0038 % but it does not hurt to allow for complex inner products 0039 % here by taking the conjugate. 0040 G(j, i) = G(i, j)'; 0041 0042 end 0043 0044 end 0045 0046 end

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