411 REFERENCE · Wikibooks
Linear Algebra and the C Language/a0of
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b1r1 + b2r2 + b3r3 = Ide (The b_r_ are projectors.)
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% A property of spectral decomposition:
%
% V1 V1' + V2 V2' + V3 V3' = Ide
% b1 r1 + b2 r2 + b3 r3 = Ide
%
% 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)
%
% 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);
b1r1 = V1*V1';
b2r2 = V2*V2';
b3r3 = V3*V3';
Ide = b1r1+b2r2+b3r3
%%