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It would be great if you could describe the steps in Layman's terms because I am not so hot in Linear Algebra.

Thanks

- Thread starter MathIdiot
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It would be great if you could describe the steps in Layman's terms because I am not so hot in Linear Algebra.

Thanks

- #2

Hurkyl

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What sort of normal forms do you know for matrices?

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Hurkyl

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Oh, ok (sorry).

They are square, they don't have any other specialized format.

They are square, they don't have any other specialized format.

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mathwonk

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then use the euclidean algorithm to decompose the space into a direct sum on each factor of which the polynomial is of form (X-c)^r.

then note that if T satisfies (X-c)^r, then it is the sum of T-cId and cId, where cID is diagonalizable, and T-cId satisfies X^r, hence is nilpotent.

done.

- #7

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Those assumptions sound fine.

However what is the euclidean algorithm?

Also, what is meant by "T satisfies (X-c)^r"?

Lastly how do you get cID? What I mean really is, how does one achieve this diagonalizable matrix?

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Euclidean algorithm, I believe, goes like this:

let f, g be polynomials in F[x] such that g does not equal 0. Then there exists uniquely determined polynomials q and r such that

f = qg + r

and r = 0 or deg r < deg g

Mathwonk, is this what is known as the primary decomposition theorem?

let f, g be polynomials in F[x] such that g does not equal 0. Then there exists uniquely determined polynomials q and r such that

f = qg + r

and r = 0 or deg r < deg g

Mathwonk, is this what is known as the primary decomposition theorem?

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