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, 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 =, where ρ 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:
  1. for some natural norm;
  2. for all natural norms;
  3. ;
  4. 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 is selected, the sequence of approximate solution vectors is generated by computing
for each k ≥ 0. For any initial vector x ∈, the sequence defined by, for each k ≥ 0 and c ≠ 0, converges to the unique solution of if and only if ρ < 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 < 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 ∈ if and only if T is semi-convergent. In this case, the splitting is called a semi-convergent splitting of A.