Recurrence relation
Stability of linear first-order matrix recurrences

In the first-order matrix difference equation

with state vector x {\displaystyle x} and transition matrix A {\displaystyle A}, x {\displaystyle x} converges asymptotically to the steady state vector x ∗ {\displaystyle x^{*}} if and only if all eigenvalues of the transition matrix A {\displaystyle A} (whether real or complex) have an absolute value which is less than 1.
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