I have found the eigenvectors from a matrix. Now I need the corresponding eigenvalues. How can I get the eigenvalue corresponding to each eigenvector in a "efficient" way for a computer?
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closed as not a real question by Andres Caicedo, martini, TMM, Thomas, J. M. Nov 25 '12 at 11:43
It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, see the FAQ.
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Well, of course, if $x$ is an eigenvector of $A$ we have $$Ax = \lambda x$$ where $\lambda$ is the eigenvalue. Thus, you can take any non-zero component of $Ax$ and divide it by the corresponding component of the vector $x$ and you will get $\lambda$. Say the $k$:th component of $Ax$ is non-zero and the rows of $A$ are $a_1^T, a_2^T, \dots, a_n^T$, so the $k$:th component of $Ax$ will be $a_k^Tx$ and we get $$\lambda = \frac{a_k^T x}{x_k}.$$ So an efficient way to calculate the eigenvalue $\lambda$, given the eigenvector $x$, is to calculate $a_k^T x$ for $k = 1, 2, \dots, n$ until you get a non-zero value, and then get $\lambda$ from the formula above. If no $a_k^T x$ is non-zero, we get $\lambda = 0$. |
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