## Appendix A  
_Some Conditions That Imply the Powers of a Square Matrix Tend to Zero_  
This subsection discusses conditions under which the powers of a square matrix converge to zero. The spectral radius of a matrix, which is determined by its eigenvalues, plays a critical role in understanding whether the powers of the matrix will converge to zero. If the spectral radius is strictly less than one, the powers of the matrix will tend to zero. However, calculating the spectral radius can be computationally expensive, so alternative sufficient conditions are often used. A sufficient condition for ensuring that the powers of a matrix converge to zero is the existence of a matrix norm such that the norm of the matrix is less than one. Two specific norms are discussed: the 1-norm and the \(\infty\)-norm.  
If **A** is a square matrix denote by \(\sigma(\textbf{A})\) the set of eigenvalues of **A**. The number \(\rho(\textbf{A})=\max\limits_{\lambda\in\sigma(\textbf{A})}|\lambda|\) is called the spectral radius of the matrix **A**. A necessary and sufficient condition that the sequence of powers of **A** be convergent to zero is that \(\rho(\textbf{A})<1\). Unfortunately, the computationally effort for finding the spectral radius is great. That is why one can use instead condition \(\rho(\textbf{A})<1\) only sufficient conditions that imply the convergence of the sequence of powers of matrix **A**. A sufficient condition that implies that the sequence of powers of a \(n\times n\) matrix **A** is convergent to zero is the existence of a norm on \(\mathbb{R}^{n}\) such that \(\|\textbf{A}\|<1\). On \(\mathbb{R}^{n}\) consider the following norms:  
* the \(l^{1}\) norm that is \(\|\textbf{x}\|_{1}=\sum_{i=1}^{n}|x_{i}|\) and
* the \(l^{\infty}\) norm that is \(\|\textbf{x}\|_{\infty}=\max\limits_{1\leq i\leq n}|x_{i}|\).  
If \(\textbf{A}=\big{(}a_{ij}\big{)}\) is a \(n\times n\) matrix and on \(\mathbb{R}^{n}\) consider the \(l^{1}\) norm then the norm of the linear application generated by the matrix A is:  
\[\|\textbf{A}\|_{1}=\max\limits_{1\leq j\leq n}\sum_{i=1}^{n}\lvert a_{ij}\rvert.\]  
In the case on \(\mathbb{R}^{n}\) we consider the \(l^{\infty}\) norm then the norm of **A** is:  
\[\|\textbf{A}\|_{\infty}=\max\limits_{1\leq i\leq n}\sum_{j=1}^{n}\lvert a_{ij}\rvert.\]  
Let \(s=\max(\|\textbf{A}\|_{1},\|\textbf{A}\|_{\infty})\).  
**Theorem A1**.: _Let \(\mathbf{A}=\big{(}a_{ij}\big{)}\) be a \(n\times n\) matrix, \(\varepsilon>0\), \(s=max(\|\mathbf{A}\|_{1},\|\mathbf{A}\|_{\infty})\), \(\mathbf{B}_{1}=\frac{1}{s}\mathbf{A}\), \(\mathbf{B}_{2}=\frac{1}{s+\varepsilon}\mathbf{A}\). Let \(Q=\{z\in\mathbb{C}:|z|=1\}\). Then, the following assertions hold:_  
1. _If_ \(\text{det}(z\mathbf{I}-\mathbf{B}_{1})\
eq 0\) _for every_ \(z\in Q\) _then the powers of matrix_ \(\mathbf{B}_{1}\) _tend to zero._
2. _If_ \(\text{det}(z\mathbf{I}-\mathbf{B}_{1})=0\) _for some_ \(z\in Q\) _then the powers of matrix_ \(\mathbf{B}_{2}\) _tend to zero._  
Here, we denoted by \(\mathbf{I}\) the unit matrix.  
Proof of Theorem A1.: In order to prove assertion (i) suppose that \(\text{det}(z\mathbf{I}-\mathbf{B}_{1})\
eq 0\) for every \(z\in Q\). This implies that matrix \(\mathbf{B}_{1}\) has no eigenvalues on the boundary of the unit disk, hence \(\rho(\mathbf{B}_{1})<1\). Thus, the powers of the matrix \(\mathbf{B}_{1}\) tend to zero.  
In order to prove assertion (ii) suppose that \(\text{det}(z\mathbf{I}-\mathbf{B}_{1})=0\) for some \(z\in Q\). Then, \(\rho(\mathbf{B}_{1})=1\). Since \(\mathbf{B}_{2}=\frac{s}{s+\varepsilon}\mathbf{B}_{1}\) it follows that \(\rho(\mathbf{B}_{2})=\frac{s}{s+\varepsilon}<1\), hence the powers of the matrix \(\mathbf{B}_{2}\) tend to zero. \(\square\)