A necessary condition for the spectrum of nonnegative symmetric 5 x 5 matrices

作者：Loewy Raphael^*; Spector Oren

来源：Linear Algebra and Its Applications, 2017, 528: 206-272.

DOI：10.1016/j.laa.2016.08.006

• Symmetric nonnegative inverse eigenvalue problem
• Extreme matrix

Let A be a nonnegative symmetric 5 x 5 matrix with eigenvalues lambda(1) >= lambda(2) >= lambda(3) >= lambda(4) >= lambda(5). We show that if Sigma(5)(i=1) lambda(i) >= 1/2 lambda(1) then lambda(3) <= Sigma(5)(i=1) lambda(i). McDonald and Neumann showed that lambda(1) + lambda(3) + lambda(4) >= 0. Let sigma = (lambda(1), lambda(2), lambda(3), lambda(4), lambda(5)) be a list of decreasing real numbers satisfying:
1. Sigma(5)(i=1) lambda(i), >= 1/2 lambda(1), 2. lambda(3) <= Sigma(5)(i=1) lambda(i), 3. lambda(1) + lambda(3) + lambda(4) >= 0, 4. the Perron property, that is lambda(1) = max(lambda epsilon sigma) vertical bar lambda vertical bar.
We show that sigma is the spectrum of a nonnegative symmetric 5 x 5 matrix. Thus, we solve the symmetric nonnegative inverse eigenvalue problem for n = 5 in a region for which a solution has not been known before.

• 出版日期2017-9-1