411 REFERENCE · Wikibooks
Linear Algebra and the C Language/a0og
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E1 b1r1 + E2 b2r2 + E3 b3r3 = A
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% A property of spectral decomposition:
%
% E1 V1 V1' + E2 V2 V2' + E3 V3 V3' = A
% E1 b1 r1 + E2 b2 r2 + E3 b3 r3 = A
%
% V (b) : The columns of the eigenvectors of A
% V'(r) : The rows of the eigenvectors of the inverse of A
% (Here the transpose: see the first example)
%
% E : The eigenvalues of A.
%
% V1V1' (b1r1) is obtained by multiplying the first column of the eigenvector
% of A by the first row of the eigenvector of the inverse of A
clear, clc
A = round(10*randn(3)); %% A matrix 3x3
A = A'*A %% A symetric matrix
% Eigenvectors, Eigenvalues
[Evectors,Evalues] = eigs(A);
V1 = Evectors(:,1);
V2 = Evectors(:,2);
V3 = Evectors(:,3);
Evalues = diag([Evalues]);
E1 = Evalues(1);
E2 = Evalues(2);
E3 = Evalues(3);
E1b1r1 = E1.*V1*V1';
E2b2r2 = E2.*V2*V2';
E3b3r3 = E3.*V3*V3';
A = E1b1r1 + E2b2r2 + E3b3r3
%%