## Reducibility and the Fundamental Theorem of Reducible Subspaces

**Point of post: **In this post we discuss the concepts covered in section 40 of Halmos, but extend it further to discuss reducibility by arbitrary (finite) number of subspaces. Moreover, we discuss what is sometimes called that ‘Fundamental Theorem of Reducible Subspaces’.

*Motivation*

In our last post we discussed the concepts of invariant subspaces, and some characterizations of them. In this post we discuss a stronger version of invariant subspaces and show how a reducible endomorphism can be canonically decomposed into the direct sum of other endomorphisms. In essence, a reducible transformation is as nice as one can reasonably expect a endomorphism to be.

*Reducible Transformations*

Let be an -space. Then, for we call *-reducible *if there exists subspaces such that

and is invariant under for . Alternatively ,we say that is *-reduced by . *

Considering our last few posts on the direct sums of matrices we have relatively little work to do now, except to notice that:

**Theorem: ***Let be an space and suppose that is -reduced by . Then,*

**Proof:** We evidently have, by prior theorem that and the rest follows by noticing that

from where the conclusion follows.

**References:**

1. Halmos, Paul R. *Finite-dimensional Vector Spaces,*. New York: Springer-Verlag, 1974. Print

[…] next theorem gives us a similar formulation for reducibility of a linear transformation in terms of projections. […]

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