# expected value of powers of a gaussian matrix

Let $$Z$$ be a fixed $$d \times d$$ matrix and let $$G$$ be a random $$d \times d$$ matrix with each entry i.i.d. $$N(0, 1)$$.

Is it true that:

$$\mathrm{Tr}(\mathbb{E}_G[ (Z^T + G^T)^\ell (Z + G)^{\ell-k-1} Z (Z+G)^k ] ) \geq 0 \:,$$

where $$\ell, k$$ are positive integers and $$\ell - k \geq 1$$?

When $$d = 1$$, this reduces to:

$$\mathbb{E}_g[ (z+g)^{2\ell - 1} z ] \geq 0 \:,$$

which is easily verified to be true, since the LHS is of the form $$a_2 z^2 + a_4 z^4 + ... + a_{2\ell} z^{2\ell}$$ with positive coefficients.