411 REFERENCE · Wikibooks
Linear Algebra and the C Language/a0oc
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The eigenvalues of the inverse of A correspond to the inverses of the eigenvalues of 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