Mathc matrices/a260

La décomposition spectral

b1r1 * b3r3 = 0; ... ... ... b1r1 * b2r2 = 0; ... ... ... b2r2 * b3r3 = 0;

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% A property of spectral decomposition:
%
%         b1r1 * b2r2 = 0
%         b1r1 * b3r3 = 0
%         b2r2 * b3r3 = 0
%
%    This simply materializes that our projectors are orthogonal to each other.
%
% 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_b2r2 = (V1*V1')*(V2*V2')
b1r1_b3r3 = (V1*V1')*(V3*V3')
b2r2_b3r3 = (V2*V2')*(V3*V3')

%%

Nous voyons une des propriétés de la décomposition spectral:

• b1r1 * b2r2 = 0
• b1r1 * b3r3 = 0
• b2r2 * b3r3 = 0

Cela matérialise simplement que nos projecteurs sont orthogonaux entre eux.