Since the concept of block matrices stems from a variety of practical applications (discretization of partial differential equations with the finite element method or finite difference methods, structured networks, intermediate transformations in numerical computations etc.) an important task is to analyse their pseudospectra. It is well known that for a fixed perturbation size, the union of pseudospectra of diagonal blocks underestimates the pseudospectrum of a full matrix. So, one is interested to provide an upper estimate by adequately changing the size of the perturbation for the pseudospectra of diagonal blocks. For the case of 2-by-2 block triangular matrix, this was optimally done in Trefethen and Embree (2005). Here we extend this result to general block matrices, and, in turn, obtain some estimates for the distance to instability of a block matrix.

由于块矩阵的概念源于多种实际应用（使用有限元法或有限差分法离散偏微分方程，结构化网络，数值计算中的中间变换等），因此重要的任务是分析其伪谱。众所周知，对于固定的摄动大小，对角线块的伪谱的并集会低估整个矩阵的伪谱。因此，有兴趣通过适当改变对角线块的伪谱的扰动大小来提供较高的估计。对于2×2块三角矩阵，这在Trefethen和Embree（2005）中进行了优化。在这里，我们将此结果扩展到一般的块矩阵，然后，