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# lyapunov_symmetric_eig

## PURPOSE Solves AX + XA = C when A = A', as a pseudo-inverse, given eig(A).

## SYNOPSIS function X = lyapunov_symmetric_eig(V, lambda, C, tol)

## DESCRIPTION ``` Solves AX + XA = C when A = A', as a pseudo-inverse, given eig(A).

function X = lyapunov_symmetric_eig(V, lambda, C)
function X = lyapunov_symmetric_eig(V, lambda, C, tol)

Same as lyapunov_symmetric(A, C, [tol]), where A is symmetric, its
eigenvalue decomposition [V, lambda] = eig(A, 'vector') is provided as
input directly, and C is a single matrix of the same size as A.

## CROSS-REFERENCE INFORMATION This function calls:
This function is called by:

## SOURCE CODE ```0001 function X = lyapunov_symmetric_eig(V, lambda, C, tol)
0002 % Solves AX + XA = C when A = A', as a pseudo-inverse, given eig(A).
0003 %
0004 % function X = lyapunov_symmetric_eig(V, lambda, C)
0005 % function X = lyapunov_symmetric_eig(V, lambda, C, tol)
0006 %
0007 % Same as lyapunov_symmetric(A, C, [tol]), where A is symmetric, its
0008 % eigenvalue decomposition [V, lambda] = eig(A, 'vector') is provided as
0009 % input directly, and C is a single matrix of the same size as A.
0010 %
0012
0013 % This file is part of Manopt: www.manopt.org.
0014 % Original author: Nicolas Boumal, Aug. 31, 2018.
0015 % Contributors:
0016 % Change log:
0017
0018     % AX + XA = C  is equivalent to DY + YD = M with
0019     % Y = V'XV, M = V'CV and D = diag(lambda).
0020     M = V'*C*V;
0021
0022     % W(i, j) = lambda(i) + lambda(j)
0023     W = bsxfun(@plus, lambda, lambda');
0024
0025     % Normally, the solution Y is simply this:
0026     Y = M ./ W;
0027
0028     % But this may involve divisions by (almost) 0 in certain places.
0029     % Thus, we go for a pseudo-inverse.
0030     absW = abs(W);
0031     if ~exist('tol', 'var') || isempty(tol)
0032         tol = numel(C)*eps(max(absW(:))); % similar to pinv tolerance
0033     end
0034     Y(absW <= tol) = 0;
0035
0036     % Undo the change of variable
0037     X = V*Y*V';
0038
0039 end```

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