(\mathbb{C})$, is now the operator norm $$\label{op_norm2}
%%\|A\|_{op} := \sup_{\|x\|=1} \|T_Ax\|\,,
%\|A\|_{op} := \sup_{\|x\|=1} \|Ax\|\,,
\|A\|:= \sup_{\|x\|=1} \|Ax\|\,,$$ where $x = (x_1,\ldots ,x_n)\in [C(\mathcal{X})]^n$ has norm $\|x\|=\sqrt{\|x_1\|_{\infty}^2 + \cdots + \|x_n\|_{\infty}^2}$ with $\|x_i\|_{\infty}= \max_{t\in \mathcal{X}}|x_i(t)|$, for $1\le i\le n$.

Since $\mathcal{A}$ is a $C^*$-algebra, all the definitions and basic facts about the spectrum remain valid for any matrix $A$ with entries in $\mathcal{A}_0$. Thus, the spectrum $\sigma(A)$ of $A\in \mathcal{A}$ is the set of all $\lambda\in \mathbb{C}$ such that $\lambda I_n - A$ is not invertible in $\mathcal{A}$. If $0\in \sigma(A)$, we will also say that $A$ is [*singular*]{}. The set $\sigma(A)$ is a nonempty compact subset of $\mathbb{C}$ completely contained in the disk of radius $\|A\|$ centered at $0$. The definition of spectral radius and Gelfand’s formula (\[GF\]) remain valid. Hermitian matrices have real spectra, skew-Hermitian matrices have purely imaginary spectra, unitary matrices have spectra contained in $S^1$, and so forth. Note, however, that it is not true in general that a normal matrix $A$ over a $C^*$-algebra can be unitarily diagonalized [@Kadison].

In general, it is difficult to estimate the spectrum of a matrix $A=[a_{ij}]$ over a $C^*$-algebra. It will be useful for what follows to introduce the 