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?
Given matrix's eigenvectors find the corresponding eigenvalues
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linear-algebra
matrices
eigenvalues-eigenvectors
1 Answers
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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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1You can also use the [Rayleigh quotient](https://en.wikipedia.org/wiki/Rayleigh_quotient). For example, hee [here](http://ergodic.ugr.es/cphys/LECCIONES/FORTRAN/power_method.pdf#page=3). – 2017-03-03