Shorted Operators

Abstract

Let A be a positive semidefinite Hermitian operator, with partitioned matrix $A = \left[ {\begin{array}{*{20}c} a & b \\ {b^ * } & c \\ \end{array} } \right]$, where a is an $s \times s$ matrix corresponding to the subspace S. Let $A^\dag $ denote the inverse of A on its range (i.e., the Moore–Penrose generalized inverse). Define $A = \left[ {\begin{array}{*{20}c} {a - bc^\dag b^ * } & 0 \\ 0 & 0 \\ \end{array} } \right]$. Then $A_S = \sup \{ D |0\leqq D\leqq A,R(D) \subset S\} $. If A is the impedance matrix of a resistive n-port network, then $A_S $ is the impedance matrix of the network obtained by shorting the last $n - s$ ports; thus we call $A_S $ a shorted operator. Since short circuits cannot increase resistance, one would expect that $(A + B)_S \geqq A_S + B_S $; this we prove. This latter formula is related to Rosenberg’s theory of decomposition of matrix measures. Parallel connections and electrical duality lead to further algebraic theorems.