The Hilbert-Schmidt Inner Product on Complex Matrix Algebras
Point of post: In this post we discuss a natural way to define an inner product on a complex matrix algebra of the form and describe some of the properties.
Of course we know that the space of of matrices over , , is an associative unital algebra with the usual definitions of scalar multiplication, matrix addition, and matrix multiplication. It may then seem fruitful to ask what kinds of inner products one can put on it. In this post we shall define a natural one, namely called the Hilbert-Schmidt inner product and derive some interesting results about it.
Hilbert-Schmidt Inner Product
Let, as usual, denote the space of all square matrices with the usual operations. We define the map
by where as usual is the conjugate transpose (i.e. if then where ). This is called the Hilbert-Schmidt inner product on . What we’d like to claim is that is an inner product on . Some of the properties, such as linearity in the first entry, and conjugate symmetry are obvious. The fact that it is positive definite is not. For this we consider the following lemma:
Lemma: Let then .
Proof: By definition, if we let we have that the general term term of is
from where the conclusion follows.
With this we are now ready to prove the main result of this post. Namely:
Theorem: The Hilbert-Schmidt inner product is an inner product .
Proof: Clearly is linear in the first column since for every and one has that
The fact that is conjugate symmetric is clear since is transpose invariant and distributive over conjugate so that
Lastly, we see from our lemma that is positive semi-definite since if
from where positive semi-definitness follows. Thus, the function satisfies all the conditions to be a complex inner product and thus the conclusion follows.
From this we can define the Hilbert-Schmidt norm, denoted , by . We claim that this is a matrix norm (i.e. it’s a norm on in the usual sense and it also satisfies ). Indeed, this follows immediately from the Cauchy-Schwarz inequality.
1. Horn, Roger A., and Charles R. Johnson. Matrix Analysis. Cambridge [u.a.: Cambridge Univ., 2006. Print.