Solving the System: How to Characterize Matrices A?

[namu]

I am confused on how to solve the following problem.

Consider the system

$\vec{x}$_t=A$\vec{x}$+$\vec{f}$

where $\vec{x}$, $\vec{f}$ are vectors of size n and A is a
constant nxn matrix. Characterize all matrices A so that for all periodic functions
$\vec{f}$ (irrespective of period) there will be at most one periodic solution.

I think we want no complex eigenvalues for the homogenous solution to have no
periodic solutions, however am not sure and am lost with the forcing function.

 

[lippyka]

Any help would be appreciated. The matrix A needs to be invertible (non-singular) for there to be at most one periodic solution. This means that the determinant of A must be non-zero and all of the eigenvalues of the matrix must be non-zero. Also, if the matrix is diagonalizable, its eigenvalues must be real and distinct.