Show that if \(A\) is a nonsingular 3 by 3 matrix having nonnegative entries, then \(A\) has a positive real eigenvalue.
Solution
Let \[ \Delta^2=\{x=(x_1,x_2,x_3)\in\mathbb R^3:x_i\ge0,\ x_1+x_2+x_3=1\} \] be the standard \(2\)-simplex. Since \(A\) has nonnegative entries, \(Ax\) has nonnegative coordinates for every \(x\in\Delta^2\). Since \(A\) is nonsingular and \(x\ne0\), we also have \(Ax\ne0\). Hence \[ s(x)=\sum_{i=1}^3(Ax)_i>0. \] Define \[ F:\Delta^2\to\Delta^2, \qquad F(x)=\frac{Ax}{s(x)}. \] This map is continuous. By the Brouwer fixed-point theorem, there exists \(x\in\Delta^2\) with \(F(x)=x\). Therefore \[ Ax=s(x)x. \] Thus \(x\ne0\) is an eigenvector of \(A\) with real eigenvalue \[ \lambda=s(x)>0. \] Hence every nonsingular \(3\times3\) matrix with nonnegative entries has a positive real eigenvalue.