Home > manopt > manifolds > euclidean > euclideancomplexfactory.m

# euclideancomplexfactory

## PURPOSE

Returns a manifold struct to optimize over complex matrices.

## SYNOPSIS

function M = euclideancomplexfactory(m, n)

## DESCRIPTION

``` Returns a manifold struct to optimize over complex matrices.

function M = euclideancomplexfactory(m)
function M = euclideancomplexfactory(m, n)
function M = euclideancomplexfactory([n1, n2, ...])

Returns M, a structure describing the vector space of complex matrices,
as a manifold for Manopt.

The complex plane is here viewed as R^2. The inner product between two
m-by-n matrices A and B is given by: real(trace(A'*B)). This choice
guides the proper definition of gradient and Hessian for this geometry.
This is not the classical Euclidean inner product for complex matrices;
it is a real inner product.

## CROSS-REFERENCE INFORMATION

This function calls:
• hashmd5 Computes the MD5 hash of input data.
• matrixlincomb Linear combination function for tangent vectors represented as matrices.
This function is called by:

## SOURCE CODE

```0001 function M = euclideancomplexfactory(m, n)
0002 % Returns a manifold struct to optimize over complex matrices.
0003 %
0004 % function M = euclideancomplexfactory(m)
0005 % function M = euclideancomplexfactory(m, n)
0006 % function M = euclideancomplexfactory([n1, n2, ...])
0007 %
0008 % Returns M, a structure describing the vector space of complex matrices,
0009 % as a manifold for Manopt.
0010 %
0011 % The complex plane is here viewed as R^2. The inner product between two
0012 % m-by-n matrices A and B is given by: real(trace(A'*B)). This choice
0013 % guides the proper definition of gradient and Hessian for this geometry.
0014 % This is not the classical Euclidean inner product for complex matrices;
0015 % it is a real inner product.
0016 %
0018
0019 % This file is part of Manopt: www.manopt.org.
0020 % Original author: Nicolas Boumal, April 7, 2015.
0021 % Contributors:
0022 % Change log:
0023 %
0024 %   Jan. 25, 2017 (NB):
0025 %       Added functionality to handle multidimensional arrays.
0026
0027     % The size can be defined using both m and n, or simply with m.
0028     % If m is a scalar, then n is implicitly 1.
0029     % This mimics the use of built-in Matlab functions such as zeros(...).
0030     if ~exist('n', 'var') || isempty(n)
0031         if numel(m) == 1
0032             n = 1;
0033         else
0034             n = [];
0035         end
0036     end
0037
0038     dimensions_vec = [m(:)', n(:)']; % We have a row vector.
0039
0040     M.size = @() dimensions_vec;
0041
0042     M.name = @() sprintf('Euclidean space C^(%s)', num2str(dimensions_vec));
0043
0044     M.dim = @() 2*prod(dimensions_vec);
0045
0046     M.inner = @(x, d1, d2) real(d1(:)'*d2(:));
0047
0048     M.norm = @(x, d) norm(d(:), 'fro');
0049
0050     M.dist = @(x, y) norm(x(:)-y(:), 'fro');
0051
0052     M.typicaldist = @() sqrt(prod(dimensions_vec));
0053
0054     M.proj = @(x, d) d;
0055
0057
0058     M.ehess2rhess = @(x, eg, eh, d) eh;
0059
0060     M.tangent = M.proj;
0061
0062     M.exp = @exp;
0063     function y = exp(x, d, t)
0064         if nargin == 3
0065             y = x + t*d;
0066         else
0067             y = x + d;
0068         end
0069     end
0070
0071     M.retr = M.exp;
0072
0073     M.log = @(x, y) y-x;
0074
0075     M.hash = @(x) ['z' hashmd5([real(x(:)) ; imag(x(:))])];
0076
0077     M.rand = @() (randn(dimensions_vec) + 1i*randn(dimensions_vec))/sqrt(2);
0078
0079     M.randvec = @randvec;
0080     function u = randvec(x) %#ok<INUSD>
0081         u = randn(dimensions_vec) + 1i*randn(dimensions_vec);
0082         u = u / norm(u(:), 'fro');
0083     end
0084
0085     M.lincomb = @matrixlincomb;
0086
0087     M.zerovec = @(x) zeros(dimensions_vec);
0088
0089     M.transp = @(x1, x2, d) d;
0090
0091     M.pairmean = @(x1, x2) .5*(x1+x2);
0092
0093     sz = prod(dimensions_vec);
0094     M.vec = @(x, u_mat) [real(u_mat(:)) ; imag(u_mat(:))];
0095     M.mat = @(x, u_vec) reshape(u_vec(1:sz), dimensions_vec) ...
0096                         + 1i*reshape(u_vec((sz+1):end), dimensions_vec);
0097     M.vecmatareisometries = @() true;
0098
0099 end```

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