Vector space decompositions
I think I'm beginning to understand the Primary Decomposition Theorem in Linear Algebra. Yay!
First, the Basic Decomposition Theorem says that if there exist orthogonal projection operators that partition identity, then the vector space can be written as the direct sum of the subspaces yielded by the projection operators (by their actions on the vector space, that is).
The Primary Decomposition Theorem says that, given a linear transformation T on the (finite) vector space (an endomorphism), then using the minimum polynomial of the transformation, we can construct a direct sum decomposition of the vector space. We do this by constructing orthogonal projection operations that partition identity. Specifically, the subspace yielded by a projection operator's action on the vector space ends up being the kernel of an irreducible distinct factor of the minimal polynomial (well, the factor with T substituted in for the variable). Now, the projection operators consists of the minimal polynomial, less one distinct factor (which is divided out), times some other stuff obtained using the GCD of .. things I don't feel like trying to describe in words. Anyway, the projection operator's subspace is the kernel of the missing factor.
Right?
First, the Basic Decomposition Theorem says that if there exist orthogonal projection operators that partition identity, then the vector space can be written as the direct sum of the subspaces yielded by the projection operators (by their actions on the vector space, that is).
The Primary Decomposition Theorem says that, given a linear transformation T on the (finite) vector space (an endomorphism), then using the minimum polynomial of the transformation, we can construct a direct sum decomposition of the vector space. We do this by constructing orthogonal projection operations that partition identity. Specifically, the subspace yielded by a projection operator's action on the vector space ends up being the kernel of an irreducible distinct factor of the minimal polynomial (well, the factor with T substituted in for the variable). Now, the projection operators consists of the minimal polynomial, less one distinct factor (which is divided out), times some other stuff obtained using the GCD of .. things I don't feel like trying to describe in words. Anyway, the projection operator's subspace is the kernel of the missing factor.
Right?