1
$\begingroup$

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}

$\endgroup$
1
$\begingroup$

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

On the other hand, products of matrices of the type you described appear in multivariate Fisher statistics, which is covered in Chapter 2.5 in a great book by Yao et. al., Large sample covariance matrices and high-dimensional data analysis. A revised edition can be found here: https://www.researchgate.net/publication/272093238_Large_Sample_Covariance_Matrices_and_High-Dimensional_Data_Analysis

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.