Real Solutions of the Equation $\Phi^t(A)=\frac1nJ_n$\protect\unboldmath
Zhang Xian, Yang Zhongpeng, Cao Chong-guang · SIAM Journal on Matrix Analysis and Applications · 2000
For nonsingular n x n matrices A, $\Phi (A)=A\circ A^{-T}$ ($\circ $ denotes the Hadamard product and A -T the inverse transpose, (A -1 ) T , of A) arises in mathematical control theory associated with chemical engineering design problems and in a matrix theoretic problem involving the relation between the diagonal entries and eigenvalues. For any positive integer t, we show that the equation $$ \Phi ^t(A)=\frac{1}{n}J_n $$ hasat least 2 t-1 and 2 t mutually $\Phi $-distinct real solutions for the case n=4 and n > 4, respectively, and that these solutions can be determined by an inverted iteration. We also prove that the above equation has only one $\Phi $-distinct real solution {\scriptsize $(\begin{array}{cc} 1&1\\[-2pt] 1&-1 \end{array})$} for $t=1$ and has no real solutions for $t\ge 2$ when n=2. The above results answer the problem that has been proposed by Johnson and Shapiro in [ SIAM J. Algebraic Discrete Methods, 7 (1986), pp. 627--644]. When does the equation $\Phi (A)=A\circ A^{-T}=\frac{1}{n}J_n$ have a real solution?