looking for answers:eigenvalues of combinations of two matrices

Question

For the two $n\times n$ matrices $A$ and $B$, if the largest eigenvalue of them is 1 and unique and the magnitude of others less than 1, then what about the new one $C:=tA+(1-t)B$, $t\in[0,1]$? Does $C$ have the largest eigenvalue 1 and unique and the magnitude of others less than 1? How to show this?

Answer 1

Try:

$$A=\begin{bmatrix} 1.1 & -1 \\ 0.06 & 0.4 \end{bmatrix} \\ B=\begin{bmatrix} 1.2 & -1 \\ 0.14 & 0.3 \end{bmatrix}$$

$A$ and $B$ each have eigenvalues of $1$ and $1/2$ but $C(t)$ has eigenvalues $\frac{1.5 \pm \sqrt{t^2/25-t/25+1/4}}{2}$ which are strictly between $1$ and $0.5$ for $t \in (0,1)$, since $\sqrt{t^2/25-t/25+1/4}<1/2$.