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So in linear algebra class we learned that a matrix $A$ is diagonalizable if it can be written in the form: $$A=PDP^{-1}$$ The useful part was that $A^k$ can be easily computed with $PD^kP^{-1}$. The diagonal entries of $D$ were simply the eigenvalues of $A$, and the corresponding eigenvectors were the columns of $P$. If you made $P$ orthogonal, $P^{-1}=P^{T}$ which is also easy.

This all makes sense, but the part I don't get is how this somehow makes finding eigenvectors easy. Our professor hinted that it involved iterative methods i.e. $\vec{x}\mapsto A\vec{x}\mapsto A^2\vec{x}...$ which is made easy with diagonalization. But this seems unintuitive because we got the diagonalization of $A$ by using the eigenvectors/eigenvalues, so what's the point of this method then? Or am I misinterpreting the use of diagonal matrices?

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I think what your professor was referring to is the power iteration algorithm for eigendecomposition.

The basic idea is to apply $ A $ repeatedly to a random vector $ \vec{v} $. If you visualise the action of $ A $ on the vector space, every application of $ A $ stretches $ \vec{v} $ more in the direction of the eigenvector with the largest eigenvalue than in the direction of the other eigenvectors. With a sufficiently large $ k $, you should be able to approximate that eigenvector as much as required. There's a visualisation of this here. Variants of this algorithm can be used to find more than just the dominant eigenvector, and are generally faster than full diagonalisation if you only want a few dominant eigenvectors.

While you don't need to perform diagonalisation in power iteration, you do need the idea of diagonalisation to show that it works. Here, we are effectively computing $ A^k\vec{v} = PD^kP^{-1}\vec{v} $. If you think of $ P $ as a coordinate transformation, the matrix $ D^k $ essentially serves to "pick out" the largest eigenvector. To be precise, as $ k $ increases, the ratio between the largest entry in $ D^k $ and any other entry increases exponentially, causing the dominant eigenvector to quickly dominate.

In other words, diagonalisation, in your case, should be thought of as a concept (specifically the concept of transforming to the eigenbasis of the matrix) rather than an algorithm.

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  • $\begingroup$ So then does diagonalization itself have any uses? Or is it simply a way of grasping why certain things work? $\endgroup$ – rcplusplus May 9 '16 at 23:59
  • $\begingroup$ Both. As you said, it allows you to find matrix powers easily if you need to, and also it provides a transformation to coordinates where the math is a lot simpler (as in the case of 3D rotational mechanics). You can also use the idea of diagonalisation to do things like power iteration (or quantum mechanics, or proving theorems) without actually diagonalising any particular matrix. Diagonalisation is really just a geometric interpretation for eigendecomposition; you can do with it whatever you want. $\endgroup$ – mark2222 May 10 '16 at 8:44
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It's easy to find eigenvalues of a diagonal matrix(i.e. if original matrix was diagonal). Here I think you have to find the diagonal matrix through change of coordinates and not by the equation of diagonizable matrix.

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