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Question about number format

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holistic
holistic il 14 Dic 2018
Commentato: holistic il 17 Dic 2018
When I calculate the eigenvectors of the following matrix:
mat=[1,2;2,1];
[V,D]=eig(mat)
I get:
V =
-0.7071 0.7071
0.7071 0.7071
However, this is not the correct answer, see Wolfram Alpha results or verify by yourself that the correct answer are the vectors:
[1,1] and [-1,1]
Could someone explain to be what went wrong here?

Risposta accettata

Steven Lord
Steven Lord il 14 Dic 2018
However, this is not the correct answer
You're assuming there is only one correct answer. That is not a valid assumption in this case.
Multiplying the eigenvector by any non-zero scalar just scales the eigenvector, and that scaled eigenvector still satisfies the equation that eigenvectors must satisfy. This makes sense if you look at the essential definition according to Wikipedia.
"In essence, an eigenvector v of a linear transformation T is a non-zero vector that, when T is applied to it, does not change direction. Applying T to the eigenvector only scales the eigenvector by the scalar value λ, called an eigenvalue."
See this example:
% Compute eigenvalues and eigenvectors
% A = magic(6)
[V, D] = eig(A)
% Does the first eigenvalue and eigenvector satisfy A*V = V*D?
shouldBeCloseToZeroVector1 = A*V(:, 1) - V(:, 1) * D(1, 1) % Close enough
% Multiple the first eigenvector by 2
twice = 2*V(:, 1);
% Does two times the first eigenvalue and eigenvector satisfy A*V = V*D?
shouldBeCloseToZeroVector2 = A*twice - twice * D(1, 1) % Also close enough
This is not a bug.
  1 Commento
holistic
holistic il 17 Dic 2018
Thank you all, I guess that slipped my mind :).

Accedi per commentare.

Più risposte (1)

Mark Sherstan
Mark Sherstan il 14 Dic 2018
Modificato: Mark Sherstan il 14 Dic 2018
The answer is correct, MATLAB is just outputting the unit vector answer.
, where i is either 1 or 2.

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