## Direct Sum of Linear Transformations and Direct Sum of Matrices (Pt. I)

**Point of post:** In this post we discuss the concepts of the direct sum of linear transformations, the direct sum of matrices, and their connection. This is all in preparation to take a broader view of section 40 in Halmos.

*Motivation*

It’s a leitmotif in mathematics to attempt to break up, when possible, objects into smaller subobjects which are simpler but still capture the essence of the entire object. In this post we show one way to do this for linear transformations. Namely, we show how given endomorphisms on spaces which are naturally isomorphic to and endomorphisms there is a natural way to ‘put them together’ to get an endomorphism on . We will then see how the structures of the individual endomorphisms affect the structure of the ‘put together one’.

*Direct Sum of Linear Transformations and Matrices*

In essence, the direct sum of a linear transformation or matrix is breaking it down into it’s smaller components. Those of you who have taken matrix analysis or some similar course will have undoubtedly crossed paths with this notation/concept when dealing with the Jordan decomposition theorem.

Let be an -space and such that and let be -spaces such that there exists isomorphisms . Then, if then we define the *direct sum *of on to be the endomorphism by the rule

where evidently is the unique representation of with .

The two special cases to consider is when and where in which case the above reduces to

and the case when we consider and is the ismomorphism

When and is not specified it will be assumed that we are dealing the the former case.

We now show some simple consequences of the above definition

**Theorem: ***Let , and be as in the definition of direct sum. Then,*

**Proof: **We first prove that

To see this we note that

Thus, it suffices to show that the set is independent. But, this follows directly from the fact that for each we have that

The conclusion follows.

**References:**

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

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