The dimension of is , not .
Do you know Cayley-Hamilton theorem?
Studying for my final in linear algebra and came across this:
Problem: Let V be the vector space of all real n x n matrices over R. Let A be an element of V, and let W be the subspace of V spanned by . Prove that .
Thoughts: The statement seems intuitive, but I'm not sure exactly what theorem/properties it results from. I know that a subspace cannot have dimension larger than its finite parent space, so . However, my professor said that I should try to prove it by having it follow from the dimension of a matrix after raising it to higher powers; i.e. not from the obvious theorem.
I read up on Cayley Hamilton, but am a little shaky as far as its implications. According to the theorem, the characteristic equation of any n x n matrix, A, satisfies , where are scalars, and is the identity matrix. But that creates, at most, equations.
Is the dimension related to the maximum number of solutions to this system? I don't see how to show that it would be at most n.
if A satisfies ,
that is, if , so that p(A) = 0, then
which means the n-th power (and thus any higher power) of A
is a linear combination of .
but this means that this set spans W, so dim(W) ≤ n.
and the characteristic polynomial of A, det(xI - A), is just such a polynomial (of degree n).
the cayley-hamilton theorem says: if p(x) = det(xI - A), then p(A) = 0.