Convergent matrix

In linear algebra, a convergent matrix is a matrix that converges to the zero matrix under matrix exponentiation.

Background
When successive powers of a matrix T become small (that is, when all of the entries of T approach zero, upon raising T to successive powers), the matrix T converges to the zero matrix. A regular splitting of a non-singular matrix A results in a convergent matrix T. A semi-convergent splitting of a matrix A results in a semi-convergent matrix T. A general iterative method converges for every initial vector if T is convergent, and under certain conditions if T is semi-convergent.

Definition
We call an n &times; n matrix T a convergent matrix if

for each i = 1, 2, ..., n and j = 1, 2, ..., n.

Example
Let
 * $$\begin{align}

& \mathbf{T} = \begin{pmatrix} \frac{1}{4} & \frac{1}{2} \\[4pt] 0 & \frac{1}{4} \end{pmatrix}. \end{align}$$ Computing successive powers of T, we obtain
 * $$\begin{align}

& \mathbf{T}^2 = \begin{pmatrix} \frac{1}{16} & \frac{1}{4} \\[4pt] 0 & \frac{1}{16} \end{pmatrix}, \quad \mathbf{T}^3 = \begin{pmatrix} \frac{1}{64} & \frac{3}{32} \\[4pt] 0 & \frac{1}{64} \end{pmatrix}, \quad \mathbf{T}^4 = \begin{pmatrix} \frac{1}{256} & \frac{1}{32} \\[4pt] 0 & \frac{1}{256} \end{pmatrix}, \quad \mathbf{T}^5 = \begin{pmatrix} \frac{1}{1024} & \frac{5}{512} \\[4pt] 0 & \frac{1}{1024} \end{pmatrix}, \end{align}$$
 * $$\begin{align}

\mathbf{T}^6 = \begin{pmatrix} \frac{1}{4096} & \frac{3}{1024} \\[4pt] 0 & \frac{1}{4096} \end{pmatrix}, \end{align}$$ and, in general,
 * $$\begin{align}

\mathbf{T}^k = \begin{pmatrix} (\frac{1}{4})^k & \frac{k}{2^{2k - 1}} \\[4pt] 0 & (\frac{1}{4})^k \end{pmatrix}. \end{align}$$ Since
 * $$ \lim_{k \to \infty} \left( \frac{1}{4} \right)^k = 0 $$

and
 * $$ \lim_{k \to \infty} \frac{k}{2^{2k - 1}} = 0, $$

T is a convergent matrix. Note that &rho;(T) = $1⁄4$, where &rho;(T) represents the spectral radius of T, since $1⁄4$ is the only eigenvalue of T.

Characterizations
Let T be an n &times; n matrix. The following properties are equivalent to T being a convergent matrix:
 * 1) $$ \lim_{k \to \infty} \| \mathbf T^k \| = 0, $$ for some natural norm;
 * 2) $$ \lim_{k \to \infty} \| \mathbf T^k \| = 0, $$ for all natural norms;
 * 3) $$ \rho( \mathbf T ) < 1 $$;
 * 4) $$ \lim_{k \to \infty} \mathbf T^k \mathbf x = \mathbf 0, $$ for every x.

Iterative methods
A general iterative method involves a process that converts the system of linear equations

into an equivalent system of the form

for some matrix T and vector c. After the initial vector x(0) is selected, the sequence of approximate solution vectors is generated by computing

for each k &ge; 0. For any initial vector x(0) &isin; $$ \mathbb{R}^n $$, the sequence $$ \lbrace \mathbf{x}^{ \left( k \right) } \rbrace _{k = 0}^{\infty} $$ defined by ($$), for each k &ge; 0 and c &ne; 0, converges to the unique solution of ($$) if and only if &rho;(T) < 1, that is, T is a convergent matrix.

Regular splitting
A matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. In the system of linear equations ($$) above, with A non-singular, the matrix A can be split, that is, written as a difference

so that ($$) can be re-written as ($$) above. The expression ($$) is a regular splitting of A if and only if B&minus;1 &ge; 0 and C &ge; 0, that is, B&minus;1 and C have only nonnegative entries. If the splitting ($$) is a regular splitting of the matrix A and A&minus;1 &ge; 0, then &rho;(T) < 1 and T is a convergent matrix. Hence the method ($$) converges.

Semi-convergent matrix
We call an n &times; n matrix T a semi-convergent matrix if the limit

exists. If A is possibly singular but ($$) is consistent, that is, b is in the range of A, then the sequence defined by ($$) converges to a solution to ($$) for every x(0) &isin; $$ \mathbb{R}^n $$ if and only if T is semi-convergent. In this case, the splitting ($$) is called a semi-convergent splitting of A.