I know the definitions of eigenvalues and eigenvectors and know how to find them. No problem. But I don' t still imagine them in my brain.
How do we concretely imagine them? What is concretely the main work which we do, while we are finding them?
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See question with excellent illustration http://math.stackexchange.com/questions/2191089/what-is-the-mechanism-of-eigenvector/2191159#comment4509286_2191159 – Widawensen Mar 29 '17 at 08:05
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As I wrote here recently, one of the motivations for these concepts is the question: are there any lines (through the origin) that remain fixed under the linear transformation $L$? For all vectors $\mathbf v$ on such such a line, linearity of $L$ requires that they satisfy $L(\mathbf v)=\lambda\mathbf v$, where $\lambda$ is some fixed scalar. The action of $L$ along this line—equivalently, in the direction of $\mathbf v$—can be characterized quite simply as scaling by the factor $\lambda$. If $L$ is diagonalizable, then its effect is a superposition of such scalings. The picture gets a bit more complicated with deficient eigenvalues and with complex eigenvalues of real matrices (the latter indicate a rotation of some sort), but the idea that an eigenspace is an invariant subspace on which $L$’s action is particularly simple remains in these more general cases.
