Direct Sum of Linear Transformations and Direct Sum of Matrices (Pt. III)
Point of post: This is a literal continuation of this post. Treat the two posts as one contiguous object.
Direct Sum of Matrices
Now that we have defined the direct sum of linear transformations we now define the direct sum of matrices and show, what we all would hope is true, that the matrix representation with respect to the direct sum of linear transformations is the direct sum of the matrix representations of the direct summands; we’ll state this more rigorously in what follows. We first define the direct sum of matrices mechanically, reverting once again to thinking of them as being elements of the algebra of square matrices.
Let be a field and (where the ‘left superscripts’ are meant to keep track of which matrix the particular is an entry in) be matrices over of size respectively. We then define the direct sum of to be the matrix
We now show how the two concepts of direct sums of linear transformations and direct sums of matrices interact. Namely:
Theorem: Let , and be as in the definition of direct sum of linear transformations, except now assume that . Also, let be bases for and suppose that for each and we have that
[where the superscript is made to tell whether or not (say, for example) is the coefficient of or . In essence, it's another index to keep straight which of the spaces we're working in]. Then,
is an ordered basis for and
Proof: The fact that is an ordered basis for is trivial since is a basis for for each (since is an isomorphism). Now to prove that is true we begin by noticing that since for each and we have that and so by definition
From where it follows that
But, it’s fairly easy to see from that the above can be rewritten as
But, upon inspection this is equal to
1. Halmos, Paul R. Finite-dimensional Vector Spaces,. New York: Springer-Verlag, 1974. Print
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