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I need to find eigenvalues/eigenvectors of different kinds of $n \times n$ matrices. For example, how would I determine these for the matrices listed below? What is the typical process? Should I always go by the route of finding eigenvalues by finding roots of characteristic polynomial and then getting eigenvectors by solving $(\mathbf{A} - \lambda \mathbf{I})\mathbf{x} = 0$?

$\begin{bmatrix} 2&0&0\\ 1&2&0\\ 0& 1 & 2 \end{bmatrix} $

$\begin{bmatrix} 4 &1 &1 &1 \\ 1&4 &1 &1 \\ 1&1 &4 &1 \\ 1& 1& 1& 4 \end{bmatrix}$

These are just examples. Typically I want to find eigenvectors of $n \times n$ matrices. If you can show me the process of finding solution of one of these matrices, that would be helpful.

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    http://en.wikipedia.org/wiki/Eigenvalue_algorithm2010-08-09
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    The first example you gave is (lower) triangular; thus the diagonal elements are eigenvalues. By hand, you can try to generate the characteristic polynomial; a computer, however, will (attempt to) reduce your matrix to a (quasi-)triangular matrix due to the ease of computing a triangular matrix's eigenvalues.2010-08-09
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    yes but question is concerning eigenvectors which is a more challenging problem.2010-08-09
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    are these not solvable by hand?2010-08-09
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    @saminny: See Qiaochu's link, in particular, [this section](http://en.wikipedia.org/wiki/Eigenvalue_algorithm#Identifying_eigenvectors) for an example of how to calculate eigenvectors. If there is something you don't understand about that article, ask a specific question.2010-08-09
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    Also this might be interesting: **Computing Eigenvalues and Eigenvectors without Determinants** [jstor.org/stable/2691340](http://www.jstor.org/stable/2691340)2011-09-13

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