v_dualdiag

V_DUALDIAG Simultaneous diagonalisation of two hermitian matrices [A,D,E]=(W,B)
 Inputs:   W,B     Two square hermitian matrices

 Outputs:  A       Diagonalizing matrix (not normally unitary or hermitian)
           D       Real diagonal matrix elements: A'*B*A=diag(D) (see note below)
           E       Real diagonal matrix elements: A'*W*A=diag(E)

 Note: At least one of W and B must be either positive definite or negative
 definite. If this is not the case, then D=A'*B*A and E=A'*W*A will be
 complex hermitian matrices rather than being vectors of real diagonal elements.

 The columns of A will be ordered so that abs(D./E) is in descending order.
 If two output arguments are given then A will be scaled to make diag(E)=I
 but, if W is singular, this will cause some elements of A to be infinite.

 If A'*B*A=diag(D) and A'*W*A=diag(E) then A'*W*A*diag(1./E)=I so A'*B*A=A'*W*A*diag(D./E)
 and hence B*A=W*A*diag(D./E) so the columns of A are the eigenvectors of W\A or
 equivalently the generalized eigenvectors of (B,W).

 Suppose we have several N-dimensional data row-vectors arising from each of C different classes of data.
 for each class, c, we can form the mean data vector m(c) and the within-class covariance matrix W(c)
 We can then form the between class covariance matrix B by taking the covariance of the mean vectors m(1), m(2), ...
 and also the averaged within-class covariance matrix W by averaging W(1), W(2), ...
 If we then take A=v_dualdiag(W,B) and postmultiply all our original data vectors by A, we obtain new
 data vectors for which the average within-class covariance matrix is the identity and for which
 the first few components contain most of the information that is useful in discriminating between classes.

0001 function [a,d,e]=v_dualdiag(w,b)
0002 %V_DUALDIAG Simultaneous diagonalisation of two hermitian matrices [A,D,E]=(W,B)
0003 % Inputs:   W,B     Two square hermitian matrices
0004 %
0005 % Outputs:  A       Diagonalizing matrix (not normally unitary or hermitian)
0006 %           D       Real diagonal matrix elements: A'*B*A=diag(D) (see note below)
0007 %           E       Real diagonal matrix elements: A'*W*A=diag(E)
0008 %
0009 % Note: At least one of W and B must be either positive definite or negative
0010 % definite. If this is not the case, then D=A'*B*A and E=A'*W*A will be
0011 % complex hermitian matrices rather than being vectors of real diagonal elements.
0012 %
0013 % The columns of A will be ordered so that abs(D./E) is in descending order.
0014 % If two output arguments are given then A will be scaled to make diag(E)=I
0015 % but, if W is singular, this will cause some elements of A to be infinite.
0016 %
0017 % If A'*B*A=diag(D) and A'*W*A=diag(E) then A'*W*A*diag(1./E)=I so A'*B*A=A'*W*A*diag(D./E)
0018 % and hence B*A=W*A*diag(D./E) so the columns of A are the eigenvectors of W\A or
0019 % equivalently the generalized eigenvectors of (B,W).
0020 %
0021 % Suppose we have several N-dimensional data row-vectors arising from each of C different classes of data.
0022 % for each class, c, we can form the mean data vector m(c) and the within-class covariance matrix W(c)
0023 % We can then form the between class covariance matrix B by taking the covariance of the mean vectors m(1), m(2), ...
0024 % and also the averaged within-class covariance matrix W by averaging W(1), W(2), ...
0025 % If we then take A=v_dualdiag(W,B) and postmultiply all our original data vectors by A, we obtain new
0026 % data vectors for which the average within-class covariance matrix is the identity and for which
0027 % the first few components contain most of the information that is useful in discriminating between classes.
0028 
0029 %      Copyright (C) Mike Brookes 1997-2013
0030 %      Version: $Id: v_dualdiag.m 10865 2018-09-21 17:22:45Z dmb $
0031 %
0032 %   VOICEBOX is a MATLAB toolbox for speech processing.
0033 %   Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html
0034 %
0035 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0036 %   This program is free software; you can redistribute it and/or modify
0037 %   it under the terms of the GNU General Public License as published by
0038 %   the Free Software Foundation; either version 2 of the License, or
0039 %   (at your option) any later version.
0040 %
0041 %   This program is distributed in the hope that it will be useful,
0042 %   but WITHOUT ANY WARRANTY; without even the implied warranty of
0043 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
0044 %   GNU General Public License for more details.
0045 %
0046 %   You can obtain a copy of the GNU General Public License from
0047 %   http://www.gnu.org/copyleft/gpl.html or by writing to
0048 %   Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA.
0049 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0050 [a,l]=eig(b+b',w+w');               % generalized eigendecomposition
0051 if isreal(l)
0052     [d,i]=sort(abs(diag(l)),'descend'); % sort by absolute value
0053     if nargout==2
0054         q=sqrt(diag(a'*w*a))'.^(-1);    % scale to make e=1
0055         a=a(:,i).*q(ones(size(w,1),1),i); % reorder and scale columns of a
0056     else
0057         a=a(:,i);                       % reorder columns of a to match eigenvalues
0058         e=real(diag(a'*w*a));
0059     end
0060     d=real(diag(a'*b*a));
0061 else
0062     d=a'*b*a;
0063     e=a'*w*a;
0064 end