Determinant of an Exponential
Point of post: In this post I prove a neat theorem from matrix analysis three different ways.
There’s a neat little theorem which for some reason, I haven’t heard of until now. I’ll prove it in three different ways. The theorem
Theorem: Let be the set of all complex matrices and consider the topology induced by the norm
Then, given a matrix such that
converges, the following formula holds:
As stated we will prove it three ways.
Proof 1: By Schur’s Theorem we have that
where is the multiset of eigenvalues, is some unitary matrix, and is an arbitrary, inconsequential number. And so,
And so, for a fixed
which, just adding entrywise, gives
but recalling that we’re treating this matrix as really being some -tuple, and we assumed that the limit exists we see that (also remembering that the matrices are “constants”) the above equals
where we’ve disregarded that each of the ‘s is evaluated at the limit, and technically is a “different” , but it turns out it won’t matter (reusing the asterisk for different values is a common occurrences in matrix analysis textbooks). But, evidently we see that the above evaluates to
But, the determinant of a triangular matrix is the product of it’s diagonal elements. So, finally
Proof 2: To cut the length of this already long post I will assume the reader is able to prove the following lemma
Lemma: Let then if is the multiset of eigenvalues of , then for any polynomial we have that is the multiset of eigenvalues for
Now, from this we see that if has the multiset of eigenvalues of then for a fixed we see that the multiset of eigenvalues for
and so (since the determinant is the product of the eigenvalues of matrix counting multiplicity)
But, notice that since
is continuous (it’s a polynomial of the -tuple’s coordinates) we see that
and so by prior discussion
But, since each of the factors in the product converges we may split the limit across multiplication to finally get
Proof 3: This is similar to Proof 1 except it doesn’t need a theorem as powerful as Schur’s, but the theorem I’m going to use might actually be a consequence…so haha I don’t know if there’s a point. Oh well..
Lemma: Let have the topology described above, then the set of all diagonalizable matrices is dense in .
Now, let , then if is the multiset of eigenvalues for we have that
for some . Thus,
And, so more simply than before, for a fixed we have that
But, as was said before (or similarly) the are “constant” and so by the way we are defining the topology we may pass the limit to the entries of the diagonal matrix. Namely, the above says
(where we’ve implicitly used that ). It follows that
agree on a dense subset of , but since both are evidently continuous it is a simple topological consequence that they must agree on all of from where the conclusion follows.