Determine whether or not the following matrices are...

Determine whether or not the following matrices are convergent: (a) $\left(\begin{array}{rr}2 & -3 \\ 3 & 2\end{array}\right)$, (b) $\left(\begin{array}{rr}.6 & -3 \\ .3 & .7\end{array}\right)$, (c) $\frac{1}{5}\left(\begin{array}{rrr}5 & -3 & -2 \\ 1 & -2 & 1 \\ 1 & -5 & 4\end{array}\right)$, (d) $\left(\begin{array}{lll}.8 & .3 & .2 \\ .1 & .2 & .6 \\ .1 & .5 & .2\end{array}\right)$.

Determine whether or not the following matrices are convergent:
(a) $\left(\begin{array}{rr}2 & -3 \\ 3 & 2\end{array}\right)$,
(b) $\left(\begin{array}{rr}.6 & -3 \\ .3 & .7\end{array}\right)$,
(c) $\frac{1}{5}\left(\begin{array}{rrr}5 & -3 & -2 \\ 1 & -2 & 1 \\ 1 & -5 & 4\end{array}\right)$,
(d) $\left(\begin{array}{lll}.8 & .3 & .2 \\ .1 & .2 & .6 \\ .1 & .5 & .2\end{array}\right)$.

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Key Concepts

Matrix Convergence

Matrix convergence generally refers to the behavior of a sequence formed by successive powers of a matrix, M, as n tends to infinity. A matrix is said to be convergent if the limit of M raised to the power n exists (often converging to the zero matrix or some idempotent matrix), and its entries stabilize as n increases.

Eigenvalue Analysis

The eigenvalues of a matrix play a critical role in determining the convergence of its powers. In particular, if all eigenvalues of a matrix lie strictly inside the unit circle (i.e., their absolute values are less than one), then the powers of the matrix will converge. If any eigenvalue has an absolute value equal to or greater than one, special additional analysis may be required.

Spectral Radius

The spectral radius of a matrix is defined as the largest absolute value of its eigenvalues. It directly determines the convergence of the matrix powers since the condition for convergence (i.e., that the matrix is contractive) is that the spectral radius is less than one. When the spectral radius is less than one, any repeated application of the matrix will eventually dampen the effects of its initial state.

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