(sp) matrix

This function checks whether a given matrix is a (sp) matrix or not.

Recently X.Xue and L.Guo has proved the necessary and sufficient condition for checking the asymptotic stability of the zero solution of the discrete system of the form
x(n+1) = Ax(n), n = 0,1,2, ....
by introducing a new class of matrices called (sp) matrices.

The LaTeX code for the definition is as follows.

\textbf{Definition:} We call $A\in s$ a (sp) matrix if there exists $m \in N $ and a sequence of subscript sets $\{I_1^{(k)}\}$, $\{I_2^{(k)}\}$, $ k = 0,1, ...,m $ from $I=\{1,2,... ,n\}$ such that
\begin{eqnarray*}
I &=& I_1^{(0)} \cup I_2^{(0)},I_1^{(0)}=
\{i:\Sigma_{j=1}^{n}a_{ij} < 1 \}, I_2^{(0)}= \{ i:\Sigma_{j=1}^{n}a_{ij} = 1 \} \\
I_1^{(k)} &=& \{i \in I_2^{(k-1)} : \exists j\in I_1^{(k-1)} \textrm{ such that } a_{ij} \neq
0 \} \\
I_2^{(k)} &=& \{i \in I_2^{(k-1)} : \forall j \in I_1^{(k-1)} \textrm{ such that } a_{ij} =
0 \}, k=1,2,...,m-1 \\
I_1^{(m)} &=& I_2^{(m-1)}\\
I_2^{(m)} &=& \phi
\end{eqnarray*}

where $I_1^{(k)}$ and $I_2^{(k )}$ , $k=0,1,2,...,m-1$ are nonempty or $I_2^{(0)} = \phi $. \\

Here $ s = \{A=(a_{ij})_{n \times n} : a_{ij} \geq 0, \Sigma_{j=1}^{n} a_{ij} \leq 1,\forall
i=1,2,...,n\}$.

If $I_2^{(0)} = \phi$, the matrix $A\in s$ is trivial (sp) matrix.

The asymptotic behavior can also be studied by spectral radius of system matrix A but often it turns out to be difficult if the size of the matrix is large. Therefore (sp) matrix condition is convenient to verify in many practical problems.

Our program checks a given matrix is a (sp) matrix or not. It displays following messages.

"True - a (sp) matrix", if the given matrix is a (sp) matrix.
"True - a trivial (sp) matrix", if the given matrix is a trivial (sp) matrix.
"False - Not a (sp) matrix", if the given matrix is not a (sp) matrix.

References:
1. X.Xue, L. Guo, A kind of nonnegative matrices and its application on the stability of discrete dynamical systems,
J. Math.Anal.Appl. (2006),doi:10.1016/j.jmaa.2006.09.053
2. R. K. George and T. P. Shah, Asymptotic Stability of Nonlinear Discrete Dynamical Systems Involving (sp) Matrix (Communicated for publication)