operator2matrix

 Forms a matrix representing a linear operator between two tangent spaces

 function [A, Bx, By] = operator2matrix(M,  x, y, F)
 function [A, Bx, By] = operator2matrix(Mx, x, y, F, [], [], My)
 function [A, Bx, By] = operator2matrix(M,  x, y, F, Bx, By)
 function [A, Bx, By] = operator2matrix(Mx, x, y, F, Bx, By, My)

 Given a manifold structure M, two points x and y on that manifold, and a
 function F encoding a linear operator from the tangent space T_x M to the
 tangent space T_y M, this tool generates two random orthonormal bases
 (one for T_x M, and one for T_y M), and forms the matrix A which
 represents the operator F in those bases. In particular, the singular
 values of A are equal to the singular values of F. If two manifold
 structures are passed, then x is a point on Mx and y is a point on My.

 If Bx and By are supplied (as cells containing orthonormal vectors in
 T_x M and T_y M respectively), then these bases are used, and the matrix
 A represents the linear operator F restricted to the span of Bx, composed
 with orthogonal projection to the span of By. Of course, if Bx and By are
 orthonormal bases of T_x M and T_y M, then this is simply a
 representation of F. Same comment if two manifolds are passed.

 See also: tangentorthobasis hessianmatrix

0001 function [A, Bx, By] = operator2matrix(Mx, x, y, F, Bx, By, My)
0002 % Forms a matrix representing a linear operator between two tangent spaces
0003 %
0004 % function [A, Bx, By] = operator2matrix(M,  x, y, F)
0005 % function [A, Bx, By] = operator2matrix(Mx, x, y, F, [], [], My)
0006 % function [A, Bx, By] = operator2matrix(M,  x, y, F, Bx, By)
0007 % function [A, Bx, By] = operator2matrix(Mx, x, y, F, Bx, By, My)
0008 %
0009 % Given a manifold structure M, two points x and y on that manifold, and a
0010 % function F encoding a linear operator from the tangent space T_x M to the
0011 % tangent space T_y M, this tool generates two random orthonormal bases
0012 % (one for T_x M, and one for T_y M), and forms the matrix A which
0013 % represents the operator F in those bases. In particular, the singular
0014 % values of A are equal to the singular values of F. If two manifold
0015 % structures are passed, then x is a point on Mx and y is a point on My.
0016 %
0017 % If Bx and By are supplied (as cells containing orthonormal vectors in
0018 % T_x M and T_y M respectively), then these bases are used, and the matrix
0019 % A represents the linear operator F restricted to the span of Bx, composed
0020 % with orthogonal projection to the span of By. Of course, if Bx and By are
0021 % orthonormal bases of T_x M and T_y M, then this is simply a
0022 % representation of F. Same comment if two manifolds are passed.
0023 %
0024 % See also: tangentorthobasis hessianmatrix
0025 
0026 % This file is part of Manopt: www.manopt.org.
0027 % Original author: Nicolas Boumal, Sep. 13, 2019.
0028 % Contributors:
0029 % Change log:
0030 
0031     % By default, the two points are on the same manifold
0032     if ~exist('My', 'var') || isempty(My)
0033         My = Mx;
0034     end
0035 
0036     if ~exist('Bx', 'var') || isempty(Bx)
0037         Bx = tangentorthobasis(Mx, x);
0038     end
0039     if ~exist('By', 'var') || isempty(By)
0040         By = tangentorthobasis(My, y);
0041     end
0042 
0043     assert(iscell(Bx) && iscell(By), 'Bx and By should be cells.');
0044 
0045     n_in = numel(Bx);
0046     n_out = numel(By);
0047     
0048     A = zeros(n_out, n_in);
0049     
0050     for k = 1 : n_in
0051         A(:, k) = tangent2vec(My, y, By, F(Bx{k}));
0052     end
0053 
0054 end