In linear algebra, a convergent matrix is a matrix that converges to the zero matrix under matrix exponentiation.
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.
We call an n × n matrix T a convergent matrix if
for each i = 1, 2, ..., n and j = 1, 2, ..., n.
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.
Let T be an n × n matrix. The following properties are equivalent to T being a convergent matrix:
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<sup>(0)</sup> is selected, the sequence of approximate solution vectors is generated by computing
for each k ≥ 0. For any initial vector x<sup>(0)</sup> ∈ , the sequence 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.
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<sup>−1</sup> ≥ 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<sup>−1</sup> ≥ 0, then ρ(T) < 1 and T is a convergent matrix. Hence the method () converges.
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<sup>(0)</sup> ∈ if and only if T is semi-convergent. In this case, the splitting () is called a semi-convergent splitting of A.