Max-Min Properties of Matrix Factor Norms

Given a set of real matrices C₀ ,C₁ , … ,Cₖ, conditions are considered under which the equality \[ {min }α _1 , … ,α _k {max }||w|| = 1 \|(C_0 + ∑i = 1^k {α _i } C_i )w\| = {max }||w|| = 1 {min }α _1 , … ,α _k \|(C_0 + ∑i = 1^k {α _i } C_i ) w\| \] holds. It is shown that if the matrices Cᵢ ,\, i = 0,1, … ,k are normal and commute with one another, then the equality holds. In particular, this implies that if Cᵢ = Aⁱ or Cᵢ = Ak - 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 $k = 1$.