# Random matrix properties

Let $$\mathbf{H}_{N,K}$$ be a random matrix whose entries are i.i.d complex Gaussian random variables with variance $$1$$. Then, we know from the law of large number that if $$N,K\rightarrow\infty$$, we have $$\frac{1}{N}\mathbf{H}^\mathrm{H}\mathbf{H}\rightarrow\mathbf{I},$$ where $$\mathbf{I}$$ is the identity matrix. Now, I have some questions regarding this large dimension analysis: Let define $$\mathbf{A}=\mathbf{H}^\mathrm{H}\mathbf{H}$$. If $$f$$ be a continues function, is it true to write? $$f(\mathbf{A})\rightarrow f(\mathbf{\mathbb{E}[\mathbf{A}]})=f(N\mathbf{I}).$$ More over, can we write as follows? $$\mathbf{H}^{\mathrm{H}}f(\mathbf{A})\mathbf{H}\rightarrow \mathbf{H}^{\mathrm{H}}f(N\mathbf{I})\mathbf{H},$$ and for special case $$f(\mathbf{A})=\mathbf{A}$$, we have? $$\mathbf{H}^{\mathrm{H}}\mathbf{A}\mathbf{H}\rightarrow N\mathbf{H}^{\mathrm{H}}\mathbf{H}\rightarrow N^2\mathbf{I}.$$

• I don't see how this would be consistent with Marchenko-Pastur Jan 21 '20 at 9:43
• Your identity matrix $\mathbf I$ must be $K\times K$ and hence cannot be the limit of anything as $K\to\infty$. Jan 21 '20 at 14:19

Let me check this for real matrices and $$f(A)={\rm tr}\,A^2$$. The $$N$$ eigenvalues $$\mu_n$$ of $$A$$ have in the limit $$N\rightarrow\infty$$ at fixed $$N/K=\lambda\leq 1$$ the Marchenko-Pastur distribution $$\rho(\mu)=\mathbb{E}\left[\sum_{n=1}^N\delta(\mu-\mu_n)\right]=N\frac{\sqrt{\lambda_+-\mu/N}\sqrt{\mu/N-\lambda_-}}{2\pi\lambda\mu},\;\;\lambda_-<\mu/N<\lambda_+,$$ with $$\lambda_\pm=(1\pm\sqrt\lambda)^2$$. The function $$f(A)$$ tends in this limit to $$f(A)\rightarrow\int_{N\lambda_-}^{N\lambda_+}\mu^2\rho(\mu)\,d\mu=N^3(1+\lambda),$$ which differs from the answer $$f(N I)=N^3$$ conjectured in the OP.