[*matricial norm*]{} of $A$ [@ostrowski; @robert], which is defined as the $n\times n$ real nonnegative matrix $$\hat A = \left[ \begin{array}{cccc}
               \|a_{11}\| & \|a_{12}\| &\ldots  & \|a_{1n}\| \\ 
               \|a_{21}\| & \|a_{22}\| & \ldots & \|a_{nn}\| \\
               \vdots  & \ddots & \ddots & \vdots \\ 
               \|a_{n1}\| & \|a_{n2}\| &\ldots  & \|a_{nn}\| 
             \end{array}
      \right] \,.$$

The following result shows that we can obtain upper bounds on the spectral radius and operator norm of a matrix $A$ over a $C^*$-algebra in terms of the more easily computed corresponding quantities for $\hat A$. As usual, the symbol $\|\cdot\|_2$ denotes the spectral norm of a matrix with real or complex entries.

\[thm\_matricial\_norm\] For any $A\in \mathcal{A}$, the following inequalities hold:

1.  $\|A\| \le \|\hat A\|_2$;

2.  $\rho (A) \le \rho (\hat A)$.

To prove the first item, observe that $$\|A\| = \sup \|T_Ax\|_{\mathscr{H}}
= \sup \left [ \sum_{i=1}^n \left  \|\sum_{j=1}^n \pi (a_{ij}) x_j\right \|^2 
\right ]^\frac{1}{2}\,,$$ where $\pi(a_{ij})$ is the Gelfand–Naimark representation of $a_{ij}\in \mathcal{A}_0$ as a bounded operator on $\cal H$, and the sup is taken over all $n$-tuples $(x_1, x_2, \ldots ,x_n)\in
{\mathscr{H}}$ with $\|x_1\|_\mathcal{H}^2 + \|x_2\|_\mathcal{H}^2 + \cdots
+ \|x_n\|_\mathcal{H}^2  = 1$.

Using the triangle inequality and the fact that the Gelfand–Naimark 