Mathc matrices/04f

Quelques propriétés des valeurs propres.

Les valeurs propres de l'inverse de A correspondent aux inverses des valeurs propres de A :

Copy/Paste into the octave window

% The eigenvalues of the inverse of A correspond to the inverses of the eigenvalues of A:

clear, clc

n = 3;

A    = round(10*randn(n));    %% A matrix nxn
A    = A'*A                   %% A symetric matrix
invA = inv(A)

% Eigenvalues
   Evalues = eigs(A)
invEvalues = eigs(invA)

% The eigenvalues of A and the eigenvalues of invA are in inverse order
printf ("Evalues(1) * invEvalues(3) =  %.f\n",Evalues(1) * invEvalues(3))
printf ("Evalues(2) * invEvalues(2) =  %.f\n",Evalues(2) * invEvalues(2))
printf ("Evalues(3) * invEvalues(1) =  %.f\n",Evalues(3) * invEvalues(1))
%%

Screen output example:

A =
   141  -163   169
  -163   334  -215
   169  -215   329

invA =
   2.5055e-02   6.8056e-03  -8.4228e-03
   6.8056e-03   7.0166e-03   1.0894e-03
  -8.4228e-03   1.0894e-03   8.0780e-03

Evalues =
   653.943
   116.788
    33.269

invEvalues =
   3.0058e-02
   8.5625e-03
   1.5292e-03

Evalues(1) * invEvalues(3) =  1
Evalues(2) * invEvalues(2) =  1
Evalues(3) * invEvalues(1) =  1