# Spectral Radius of Random Matrix with Column-wise Variance

Consider a random matrix with mean-zero, independent elements with variance defined column-wise: $$M_{ij}$$ with $$\mathbb{E}[M_{ij}^2]=\frac{\sigma_j^2}{N}$$

and assume that the average the variances is $$\sigma^2$$, say via a doubly-stochastic process with ($$\mathbb{E}[\sigma_j^2]=\sigma^2$$).

I know from the literature on chaotic dynamics in recurrent neural networks with asymmetric connections that the spectral radius of $$M_{ij}$$ is $$\sigma$$. (The curious can see the context below)

I would like to find a proof or a citation in the random matrix literature. Any leads?

(The same question went unanswered on this forum 7 years ago: Random matrix with non-identical variances)

Alternate Formulation: Define $$\mathbf{M}=\mathbf{J}\boldsymbol{\Sigma}$$, as the matrix product of a random Gaussian i.i.d. matrix, $$J_{ij}~\mathcal{N}(0,\frac{\sigma^2}{N})$$, times a diagonal non-negative matrix, $$\boldsymbol{\Sigma}$$, with mean element equal $$1$$. If we consider $$\boldsymbol{\Sigma}$$ to be deterministic with some profile, does that help?

.....

The situation comes up often in the non-linear dynamics of neural networks with random asymmetric connections. In particular consider the dynamics defined as

$$\dot{h_i} = -h_i + \sum_j J_{ij} \phi (h_j)$$ where J_{ij} is mean-zero Gaussian i.i.d. with variance $$\frac{g^2}{N}$$, and $$\phi$$ is a sigmoidal non-linearity, say for concreteness $$\phi(h)=\tanh(h)$$

Now, for $$g>1$$ one can show that there are non-zero fixed points, $$h_i^*$$. And when one comes to study the local stability of these fixed points one writes the linearized dynamics: $$\dot{\delta h}_i=-\delta h_i + \sum_j M_{ij} \delta h_j$$ where $$M_{ij} \equiv J_{ij} \phi'(h_j^*)$$

The fixed point itself $$h_j^*$$ is assumed Gaussian and independent of $$J_{ij}$$, and therefore $$M_{ij}$$ has the form described above with $$\mathbb{E}[M_{ij}^2] = \frac{g^2}{N}\mathbb{E}[(\phi'(h_j))^2]$$.

Such a fixed point is shown in the literature to be unstable when $$g\sqrt{\mathbb{E}[(\phi'(h_j))^2]}>1$$, i.e. the spectral radius of $$M_{ij}$$ is given by $$g\sqrt{\mathbb{E}[(\phi'(h_j))^2]}$$.

See Kadmon, Sompolinsky PRX 2015 among many others.