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I am familiar to a method of diagonalizing an nxn-matrix which fulfills the following condition:

the sum of the dimensions of the eigenspaces is equal to n.

As to the algorithm itself, it says:

1. Find the characteristic polynomial.

2. Find the roots of the characteristic polynomial.

3. Let the eigenvectors [itex]v_{i}[/itex] be the column vectors of some matrix S.

4. Let the eigenvalues [itex]\lambda_{i}[/itex] be the elements of some diagonal matrix, ordered to CORRESPOND the order of the eigenvectors in S.

5. Our Diagonalization of A should be:

[itex]A = S \cdot A \cdot S^{-1} = (v_{1}...v_ {i}...v_ {n}) \cdot (\lambda_{1}...\lambda_{i}...\lambda_{n}) \cdot S^{-1}[/itex]

My question is: how do I find at least one such matrix A Corresponding to somerandomlycreated polynomial of degree [itex]m[/itex] withinteger roots? If it is too difficult to solve this for an arbitrary [itex] m [/itex], that's okay. But let's say for [itex] m = 5[/itex]? Or for the much simpler case of [itex] m = 2[/itex]?

Is this somehow related to the quadratic form?

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# Eigenvalues and characteristic polynomials

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