A limit theorem for monotone matrix functions (Q1318225)

The author considers the limit problem for symmetric matrix functions and proves that if \(Q(t)= U(t) X^{-1}(t)\) is a symmetric \(m\times m\) matrix function which decreases on some interval \((0, \varepsilon)\) and \(U(t)\to U\), \(X(t)\to X\) as \(t\to 0^ +\) with \(\text{rank}(U^ T, X^ T)= m\), then \(\lim_{t\to 0^ +} X^ T Q(t) X= U^ T X\) and \(\lim_{t\to 0^ +} c^ T Q(t) c= \infty\) for all \(c \overline\in \text{ Im}(X)\).