Can a Matrix Be Expressed as a Sum of Diagonalizable and Nilpotent Matrices?

[MathIdiot]

How do you write a matrix as a sum of a diagonalizable matrix and a nilpotent matrix?

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

Thanks

 

[Hurkyl]

What sort of normal forms do you know for matrices?

 

[MathIdiot]

I'm not sure what you mean. The only ideas about "normal" I know is normalizing a vector. Normalizing a basis of vectors as well as normalizing a basis of orthogonal vectors to get an orthonormal basis. And orthogonal matrices. Is this at all what your implying?

 

[Hurkyl]

No; I'm using "normal form" to describe the category of things like row echelon form or diagonal form -- methods for representing matrices in some sort of specialized format which is easy to manipulate.

 

[MathIdiot]

Oh, ok (sorry).
They are square, they don't have any other specialized format.

 

[mathwonk]

it is not true in general,so first assume the field is algebraically closed. then assume you know that every linear map satisfies some minimal polynomial

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.

 

[MathIdiot]

Ok, I hope you can clarify a few things from your last statement.
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?

 

[The|M|onster]

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