Mathematical aspects of the relative gain array ( AΦHA —T )

For nonsingular n-by-n matrices A, we investigate the map \[ A \to \Phi ( A ) \equiv A \circ ( A^{ - 1} )^T \] in which $ \circ $ denotes the Hadamard (entry-wise) product. The matrix $\Phi ( A )$ arises in mathematical control theory in chemical engineering design problems, where it is known as the relative gain array, and also in a matrix theoretic problem involving the relation between the diagonal entries and eigenvalues. We first give several elementary properties of $\Phi $ and show that the iterates $\Phi ^k ( A )$ converge to I when A is either positive definite or an H-matrix. We then discuss, with examples and partial results, several unsolved problems associated with $\Phi $. These include the range of $\Phi $, inverse images of elements in the range of $\Phi $, fixed points of $\Phi $, etc.