On the Inertia of Intervals of Matrices

The inertia of intervals and lines of matrices is investigated. For complex $n \times n$ matrices A and B it is shown that, under mild nonsingularity conditions, $A + tB$ changes inertia at no more than $n^2 $ real values of t. Conditions are given for the constancy of the inertia of $A + tB$, where t lies in a real interval. These conditions generalize and organize some known results.