Convergent matrix

In numerical 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" × "n" matrix T a convergent matrix if
for each "i" = 1, 2, ..., "n" and "j" = 1, 2, ..., "n".
Example.
Let
Computing successive powers of T, we obtain
and, in general,
Since
and
T is a convergent matrix. Note that "ρ"(T) = , where "ρ"(T) represents the spectral radius of T, since is the only eigenvalue of T.
Characterizations.
Let T be an "n" × "n" matrix. The following properties are equivalent to T being a convergent matrix:
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" ≥ 0. For any initial vector x(0) ∈ formula_11, the sequence formula_12 defined by (), for each "k" ≥ 0 and c ≠ 0, converges to the unique solution of () if and only if "ρ"(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−1 ≥ 0 and C ≥ 0, that is, and C have only nonnegative entries. If the splitting () is a regular splitting of the matrix A and A−1 ≥ 0, then "ρ"(T) < 1 and T is a convergent matrix. Hence the method () converges.
Semi-convergent matrix.
We call an "n" × "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) ∈ formula_11 if and only if T is semi-convergent. In this case, the splitting () is called a semi-convergent splitting of A.