Thread: Matrix transformation

How do I start this question? Do I let P equal the 2x2 matrix abcd?

I would go about it by showing that A doesn't have 2 linearly independent eigenvectors. Then there's a theorem that states you can't diagonalize an n x n matrix unless you have n linearly independent eigenvectors. Make sense?

I don't know what an eigenvector is. I think diagonalising is a method of finding an inverse matrix but I'm not familiar with that either. I'll look into these things.

If you don't know what an eigenvector is, then I suppose you could set



Then you can compute the "inverse", if it exists, as



Multiply out and show that if you set it equal to



for real then there is no solution. Does that make sense?

Yes, I have already multiplied it out. It is quite lengthy to do.

Ah. So I get



Is that what you get? If so, you can tell, almost by inspection, that you can't make this into the diagonal matrix specified above. Why is that?

Yes I got the same. If c and d equal 0 then k1 and k2 would be 0 since they only comprise of symbols multiplied by c or d.

Well, that would only indicate that is a possibility. What about that denominator out front?

(-c^2)/(ad-bc) = (d^2)/(ad-bc) = 0

Multiply all by ad-bc then (-c^2) = (d^2). Any valued squared is positive so c and d can only be 0.

I'm not sure you're seeing the right thing here. If , then what must ? And what is relative to ?

I see, P must be a null matrix.

I don't exactly know what you mean by "null matrix". What is that?

I see what you mean now. The discriminant of P is 0 hence P is a singular matrix. A null matrix or zero matrix has every element 0.

Well, the correct term is "determinant". But you have the idea. P is technically a null matrix as well. But what's important for your problem is that it is singular. So I'd say you're pretty much done, then.