Given a sequence of vectors \left(v_n \right)_n with v_{n+1} =A \cdot v_n and
A =
\begin{pmatrix} A & B\\
C & D \end{pmatrix}.
Determine the entry {\color{red}y} such that the sequence with initial vector {\color{orange}v_0} =
\begin{pmatrix} X \\
{\color{red}y} \end{pmatrix}
converges to the zero vector.
\color{red} y
=
2 * X
The matrix has an eigenvalue {\color{blue}\lambda} with |{\color{blue}\lambda}| < 1 .
It's {\color{blue}\lambda}= fractionReduce(pow(-1,SN)*N,N+K).
We are looking for {\color{red}y} with v_1 = A \cdot {\color{orange}v_0} =
{\color{blue}\lambda} {\color{orange}v_0},
i.e. the initial vector {\color{orange}v_0} is an eigenvector for the eigenvalue {\color{blue}\lambda} =
fractionReduce(pow(-1,SN)*N,N+K).
Then further v_n = A^n \cdot {\color{orange}v_0} =
{\color{blue}\lambda}^n {\color{orange}v_0} and with |{\color{blue}\lambda}| < 1
the sequence of numbers converges \left({\color{blue}\lambda}^n \right)_n to zero.
We calculate \begin{pmatrix} A & B\\
C & D \end{pmatrix} \cdot \begin{pmatrix} X \\
{\color{red}y} \end{pmatrix} =\begin{pmatrix}
negParens(A) \cdot negParens(X) + negParens(B) \cdot {\color{red}y} \\
negParens(C) \cdot negParens(X) + negParens(D) \cdot {\color{red}y}
\end{pmatrix} =
\begin{pmatrix}
negParens(fractionReduce(X*(pow(-1,SN)*N-2*BZ),N+K)) + negParens(B) \cdot {\color{red}y} \\
negParens(fractionReduce(X*(2*(pow(-1,SN)*N-2*BZ)+4*(BZ)-2*D*(N+K)),N+K)) +
negParens(D) \cdot {\color{red}y}
\end{pmatrix}
and try to find {\color{red}y} with
\begin{pmatrix}
negParens(fractionReduce(X*(pow(-1,SN)*N-2*BZ),N+K)) + negParens(B) \cdot {\color{red}y} \\
negParens(fractionReduce(X*(2*(pow(-1,SN)*N-2*BZ)+4*(BZ)-2*D*(N+K)),N+K)) +
negParens(D) \cdot {\color{red}y}
\end{pmatrix} = {\color{blue}negParens(fractionReduce(pow(-1,SN)*N,N+K))} \cdot
\begin{pmatrix} X \\
{\color{red}y} \end{pmatrix}.
Using the first row
negParens(fractionReduce(X*(pow(-1,SN)*N-2*BZ),N+K)) + negParens(B) \cdot {\color{red}y}
= fractionReduce(X*pow(-1,SN)*N,N+K) we get
{\color{red}y} = 2*X .
Thus, for n \to \infty we get for the coordinates of the vectors
v_n =
\begin{pmatrix} \left(fractionReduce(pow(-1,SN)*N,N+K)\right)^n \cdot negParens(X) \\
\left(fractionReduce(pow(-1,SN)*N,N+K)\right)^n \cdot negParens(2*X)\end{pmatrix}
\to \begin{pmatrix} 0 \\
0 \end{pmatrix}.