I have a question about marginal stability of a system:
\begin{equation}
 \mathbf{x}[k] = \mathbf{A}\mathbf{x}[k-1]
\end{equation}
I would adapt the definition of marginal stability from this [question](https://math.stackexchange.com/questions/1525110/relationship-between-bibo-marginal-and-asymptotic-stability-statements) to the above discrete system. The system is marginally stable if the signal $\mathbf{x}[k]$ is bounded, i.e.:
\begin{equation}
\lim\_{k\rightarrow\infty} \mathbf{x}[k] < M < \infty
\end{equation}
I have trouble finding the correct book reference.


Most of the references I have found talk about asymptotic stability, and state that spectral radius of matrix should be $\rho(\mathbf{A}) < 1$.
If at least one eigenvalue of matrix $\mathbf{A}$ is outside unit circle, the above system is unstable.


I have read in few references that multiple same eigenvalues result in the unstable matrix.
However, I don't think this is the case for the unit matrix:
\begin{equation}
 \mathbf{A} = \begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}
\end{equation}
In this case the matrix has two same eigenvalues with value $1$, and matrix is marginally stable.
What confuses me is that you can have the following matrix:
\begin{equation}
 \mathbf{A} = \begin{bmatrix}
2 & -1\\
1 & 0
\end{bmatrix}
\end{equation}
 with same eigenvalues, but this system is unstable.


Can marginal stability be characterized by the location of eigenvalues? How can I determine whether the system above is stable by analyzing the matrix $\mathbf{A}$. If possible, could you provide a reference?



Marginal stability of discrete linear time-invariant system
The matrix $\mathbf{A}$ is similar to the matrix
$\begin{bmatrix}
1 & 1\\
0 & 1
\end{bmatrix}$, which is not marginally stable. Hence the original matrix is not stable. 


In general, one can apply Jordan decomposition on $\mathbf{A}$, and it is marginally stable if and only if there are no eigenvalues larger than $1$ and there are no nontrivial jordan block with diagonal $1$.


