# Large scale analysis of matrix multiplications

Let $$\mathbf{A}_{m\times n}$$ and $$\mathbf{B}_{m\times n}$$ be two random i.i.d matrices with zero mean and unit variance. Then, are the following large-scale analysis true (m,n go to infinity with fixed ratio)? \begin{align} \mathbf{A}^\mathrm{H}(\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}=\frac{1}{m}\mathbf{A}^\mathrm{H}(\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}, \end{align} as $$\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H}\rightarrow\mathbf{I}_m$$, where $$\mathbf{I}_m$$ is an identity matrix, we have \begin{align} \frac{1}{m}\mathbf{A}^\mathrm{H}(\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}&\rightarrow\frac{1}{m}\mathbf{A}^\mathrm{H}\mathbf{A}\\ &=\frac{n}{m}\frac{1}{n}\mathbf{A}^\mathrm{H}\mathbf{A}\\ &\rightarrow \frac{n}{m}\mathbf{I}_n. \end{align}

If $$m,n \to \infty$$ with $$\frac{m}n \to \lambda \in (0,\infty)$$ (let's suppose that $$n$$ depends on $$m \in\mathbb{N}$$), and if you are saying that each $$\mathbf{B} = \mathbf{B}_m$$ is an $$m\times n$$ random matrix whose entries are i.i.d. drawn from a fixed (across $$n$$) probability distribution $$\mu$$ with zero mean and unit variance, then the behavior of $$\frac1m \mathbf{B}\mathbf{B}^\dagger$$ is known to be potentially very far from $$\mathbf{I}_m$$, and even much worse its inverse. This is the content of the Marcenko-Pastur law, https://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution