I'm just going through some questions about stability from my course. The questions relate to determining for which $F$ the matrix $A−BF$ is asymptotically stable.
The question which I am confused is:
Determine for which F the matrix A-BF is stable, where
A=\begin{array}{ccc} -1 & 0 \\ 0 & 2 \\ \end{array}
and
B=\begin{array}{ccc} 0 \\ 1 \\ \end{array}
In the lecture notes and some of the other questions on problem sheets the lecturer always considers for which values of F, $det(A-BF-\lambda I)$ is stable. But in the question above he only considers $det(A-BF)$ and I am unsure why - as it gives a different result than when considering $det(A-BF-\lambda I)$.
The closed loop system matrix is $$ A-BF=\begin{bmatrix}-1 & 0\\0 & 2\end{bmatrix}-\begin{bmatrix}0\\1\end{bmatrix}\begin{bmatrix}F_1 & F_2\end{bmatrix}=\begin{bmatrix}-1 & 0\\-F_1 & 2-F_2\end{bmatrix}. $$ It has two eigenvalues $-1$ and $2-F_2$ (easy to see as the matrix is triangular, so the eigenvalues are on the main diagonal). To be stable both must have negative real parts. The first one is no problem, the second one should satisfy $$ 2-F_2<0\quad\Leftrightarrow\quad F_2>2. $$