Shorted Operators

Let A be a positive semidefinite Hermitian operator, with partitioned matrix A = [ array*20c a & b \\ b^ * & c \\ array ], where a is an s × s matrix corresponding to the subspace S. Let A^ denote the inverse of A on its range (i.e., the Moore–Penrose generalized inverse). Define A = [ array*20c a - bc^ b^ * & 0 \\ 0 & 0 \\ array ]. Then AS = \ D |0 D A,R(D) ⊂ S\. If A is the impedance matrix of a resistive n-port network, then AS is the impedance matrix of the network obtained by shorting the last $n - s$ ports; thus we call AS a shorted operator. Since short circuits cannot increase resistance, one would expect that (A + B)S AS + BS; 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.