Proving V as a Direct Sum: Showing T Can Be Represented by an nxn Matrix

[Treadstone 71]

"Suppose that T : V -> V is a linear transformation of vector spaces over
R whose minimal polynomial has no multiple roots. Show that V can be
expressed as a direct sum

V = V1 + V2 + · · · + Vt

of T-stable subspaces of dimensions at most 2. Show that, relative to a suitable basis, T can be represented by an n × n matrix with at most 2n non-zero entries, where n := dim(V)."

Our professor is a little behind in lectures, but our assignments are still rolling full speed ahead. I'm not sure where to start. Can someone give me a hint?

 

[Hurkyl]

Hrm. My first thought is the fundamental theorem of algebra.

 

[matt grime]

If it were over C then of course you can diagonalize, but over R you can't. Remember roots will come in complex conjugate pairs because the minimal poly has real coefficients.

Put that together...

 

[Treadstone 71]

You're saying even if there are no real roots, I can somehow use a 2x2 matrix to "represent" a complex root in a real vector space? Like

 0 1
-1 0

 

[matt grime]

Not only am I saying that but so is the question.

 

[Treadstone 71]

I can't get it right. What if the minimal poly has a single root only, where dim V>1?

 

[matt grime]

If it is a single root then it is necessarily real, since the characteristic poly is over R. All roots occur as complex conjugate pairs. In this case the matirx is diagonalizable, over R.

 

[Treadstone 71]

How can you conclude from the fact that it has a single root that it's diagonalizable over R?

 

[matt grime]

Perhaps I was leaping to the conclusion that you meant single as in without multiplicity. If your 'minimal poly' of M has a single root, ie is m(x)=x-t, then M=tI. Since you have has a hypothesis that your minimal polynomial has no repeated roots we are in this situtation. Sorry if I didn't make it clear to you that I was using the hypotheses of the question. It is certainly not true that just because a matrix has one e-value it is diagonalizable, never mind over R or any other field. But we have the extra assumption here that we *have* to use.

The minimal poly is real with no repeated roots, ie it is of the form

m(x)=(x-a)(x-b)..(x-c)(x^2+dx+e)..(x^2+fx+g)

where all numbers are real, no repeated roots and all quadratics have non-real roots.

 

[Treadstone 71]

So by "not multiple" the question really means "non-repeating"? I've been doing it assuming that it was either 1 or no real roots.

 

[matt grime]

Erm, just look up mutliple roots anywhere at all. I do not even know what you mean by 'one or no real roots'. Multiplicity of roots is a well defined concept if you ask me.