Notes on the Norm Estimates for the Sum of Two Matrices

This is a lecture note of my joint work with Chi-Kwong Li concerning various results on the norm structure of n × n matrices (as Hilbert-space operators). The main result says that the triangle inequality serves as the ultimate norm estimate for the upper bounds of summation of two matrices. In the case of summation of two normal matrices, the result turns out to be a norm estimate in terms of the spectral variation for normal matrices.