Max-Min Properties of Matrix Factor Norms
Anne Greenbaum, Leonid I Gurvits · SIAM Journal on Scientific Computing · 1994
Given a set of real matrices $C_0 ,C_1 , \ldots ,C_k $, conditions are considered under which the equality \[ \mathop {\min }\limits_{\alpha _1 , \ldots ,\alpha _k } \mathop {\max }\limits_{||w|| = 1} \left\|\left(C_0 + \sum_{i = 1}^k {\alpha _i } C_i \right)w\right\| = \mathop {\max }\limits_{||w|| = 1} \mathop {\min }\limits_{\alpha _1 , \ldots ,\alpha _k } \left\|\left(C_0 + \sum_{i = 1}^k {\alpha _i } C_i \right) w\right\| \] holds. It is shown that if the matrices $C_i ,\, i = 0,1, \ldots ,k$ are normal and commute with one another, then the equality holds. In particular, this implies that if $C_i = A^i $ or $C_i = A^{k - i} $, where A is a normal matrix, then the equality holds. An example is given to show that the equality may fail for noncommuting matrices, when $k > 1$. It is shown that the equality holds for arbitrary matrices if $k = 1$.