Sturm's Theorem for Min matrices

Abstract

In the present paper, we study Min matrix $ \mathcal{A}_{min} = \left[a_{min\left(i, j\right)}\right]_{i, j = 1}^n $, where $ a_s $'s are the elements of a real sequence $ \left\lbrace a_s\right\rbrace $. We first obtain a recurrence relation for the characteristic polynomial for matrix $ \mathcal{A}_{min} $, and some relations between the coefficients of its characteristic polynomial. Next, we show that the sequence of the characteristic polynomials of the $ i \times i \left(i \leq n\right) $ Min matrices satisfies the Sturm sequence properties according to different required conditions of the sequence $ \left\lbrace a_s\right\rbrace $. Using Sturm's Theorem, we get some results about the eigenvalues, such as the number of eigenvalues in an interval. Thus, we obtain the number of positive and negative eigenvalues of Min matrix $ \mathcal{A}_{min} $. Finally, we give an example to illustrate our results.

Keywords

• sturm's theorem
• min-max matrices
• characteristic polynomial
• eigenvalue