3.1 The Gram matrix
Let \((\Omega , \mu )\) be a measure space, let \(\iota \) be a finite set, let \(a : \Omega \to \mathbb {C}^{\iota }\) be square-integrable in every coordinate, and write \(G(a) = \int _\Omega a(x) a(x)^* \, d\mu (x)\).
(a) For every matrix \(Q \in \mathbb {C}^{\iota \times \iota }\),
\[ \int _\Omega a(x)^* Q \, a(x) \, d\mu (x) = \operatorname {Tr}\bigl( Q \, G(a) \bigr). \]
(b) For every matrix \(S \in \mathbb {C}^{\iota \times \iota }\),
\[ G(S a) = S \, G(a) \, S^* . \]
Proof
(a) Expand both sides as a double sum over \(\iota \) and exchange the finite sum with the integral. The integrand is integrable by Cauchy–Schwarz for \(L_2\)-functions.
(b) Expand both sides entrywise and exchange the finite sums with the integral.