# Expected norm of a product of Gaussian matrices

Suppose $$C_n$$ is a product of $$n$$ $$d\times d$$ matrices with IID entries coming from standard normal. The following appears to be true. Is there an elementary proof?

$$E[\|C_n\|_F^2]=d^{n+1}$$

This follows from discussion on math.SE on the moment method, but unclear how to adapt it to this, since the moment method requires fixing $$n$$.

This follows from the fact that $$\mathbb{E}[A^\dagger A]=d I$$ (with $$A^\dagger$$ the conjugate transpose of $$A$$ and $$I$$ the $$d\times d$$ identity matrix). Hence $$\mathbb{E}[\|C_n\|_F^2]=\operatorname{tr}\mathbb{E}[A_1A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger A_1^\dagger]$$ $$=\operatorname{tr}\mathbb{E}[A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger]\mathbb{E}[ A_1^\dagger A_1]$$ $$=d\operatorname{tr}\mathbb{E}[A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger]$$ $$=d\operatorname{tr}\mathbb{E}[A_3\cdots A_nA_n^\dagger\cdots A_3^\dagger]\mathbb{E}[ A_2^\dagger A_2]$$ $$=d^2\operatorname{tr}\mathbb{E}[A_3\cdots A_nA_n^\dagger\cdots A_3^\dagger]$$ $$=d^{n-1}\operatorname{tr}\mathbb{E}[A_n^\dagger A]=d^n\operatorname{tr} I=d^{n+1}.$$