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I am reading a paper on covariance matrix estimation, and in this paper is introduced a class of covariance matrices:

\begin{equation} U(q, c_0(p),M)=\{\Sigma: \sigma_{ii}\leq M,\quad \max_j\sum_{j=1}^p|\sigma_{ij}|^q\leq c_0(p),\forall i\}, \end{equation} where $0\leq q<1$ and $p$ is a dimension of covariance matrix.

In article is said that \begin{equation} \lambda_{max}(\Sigma)\leq\max_i\sum_{j=1}^p|\sigma_{ij}|\leq M^{1-q}c_0(p). \end{equation} The first inequality is clear, since $\Sigma$ is symmetric matrix. I tried to bound this norm by product of $\max_{i,j}|\sigma_{ij}|$ (which is bounded by $M$) and number of non-zero elements in row but I cannot obtain expected result.

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  • $\begingroup$ If $\max_{i,j}|\sigma_{ij}|\le M$, then $|\sigma_{ij}|\le M^{1-q}|\sigma_{ij}|^q$, whence the desired inequality follows. $\endgroup$
    – Iosif Pinelis
    Commented Oct 30 at 17:12
  • $\begingroup$ Oh, I actually missed a few transformations. Thank you. I have one more question: this class is further constrained by a condition $0<\epsilon_0\leq\lambda_{min}(\Sigma)$. Does this mean in this case that because the smallest eigenvalue isn’t close to zero, we won’t have any issues with inverting the matrix? $\endgroup$
    – spenziak
    Commented Oct 30 at 17:42

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