# Convergent matrix

In the mathematical discipline of numerical linear algebra, 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 is called a convergent 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

${\displaystyle \lim _{k\to \infty }({\mathbf {T}}^{k})_{ij}={\mathbf {0}},\quad (1)}$

for each i = 1, 2, ..., n and j = 1, 2, ..., n.[1][2][3]

## Example

Let

{\displaystyle {\begin{aligned}&\mathbf {T} ={\begin{pmatrix}{\frac {1}{4}}&{\frac {1}{2}}\\[4pt]0&{\frac {1}{4}}\end{pmatrix}}.\end{aligned}}}

Computing successive powers of T, we obtain

{\displaystyle {\begin{aligned}&\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{aligned}}}
{\displaystyle {\begin{aligned}\mathbf {T} ^{6}={\begin{pmatrix}{\frac {1}{4096}}&{\frac {3}{1024}}\\[4pt]0&{\frac {1}{4096}}\end{pmatrix}},\end{aligned}}}

and, in general,

{\displaystyle {\begin{aligned}\mathbf {T} ^{k}={\begin{pmatrix}({\frac {1}{4}})^{k}&{\frac {k}{2^{2k-1}}}\\[4pt]0&({\frac {1}{4}})^{k}\end{pmatrix}}.\end{aligned}}}

Since

${\displaystyle \lim _{k\to \infty }\left({\frac {1}{4}}\right)^{k}=0}$

and

${\displaystyle \lim _{k\to \infty }{\frac {k}{2^{2k-1}}}=0,}$

T is a convergent matrix. Note that ρ(T) = {{ safesubst:#invoke:Unsubst||$B=1/4}}, where ρ(T) represents the spectral radius of T, since {{ safesubst:#invoke:Unsubst||$B=1/4}} 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

{{#invoke:main|main}} A general iterative method involves a process that converts the system of linear equations

${\displaystyle {\mathbf {Ax}}={\mathbf {b}}\quad (2)}$

into an equivalent system of the form

${\displaystyle {\mathbf {x}}={\mathbf {Tx}}+{\mathbf {c}}\quad (3)}$

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

${\displaystyle {\mathbf {x}}^{(k+1)}={\mathbf {Tx}}^{(k)}+{\mathbf {c}}\quad (4)}$

for each k ≥ 0.[8][9] For any initial vector x(0)${\displaystyle \mathbb {R} ^{n}}$, the sequence ${\displaystyle \lbrace {\mathbf {x}}^{\left(k\right)}\rbrace _{k=0}^{\infty }}$ defined by (4), for each k ≥ 0 and c ≠ 0, converges to the unique solution of (3) if and only if ρ(T) < 1, i.e., T is a convergent matrix.[10][11]

## Regular splitting

{{#invoke:main|main}} A matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. In the system of linear equations (2) above, with A non-singular, the matrix A can be split, i.e., written as a difference

${\displaystyle {\mathbf {A}}={\mathbf {B}}-{\mathbf {C}}\quad (5)}$

so that (2) can be re-written as (4) above. The expression (5) is a regular splitting of A if and only if B−10 and C0, i.e., B−1 and C have only nonnegative entries. If the splitting (5) is a regular splitting of the matrix A and A−10, then ρ(T) < 1 and T is a convergent matrix. Hence the method (4) converges.[12][13]

## Semi-convergent matrix

We call an n × n matrix T a semi-convergent matrix if the limit

${\displaystyle \lim _{k\to \infty }{\mathbf {T}}^{k}\quad (6)}$

exists.[14] If A is possibly singular but (2) is consistent, i.e., b is in the range of A, then the sequence defined by (4) converges to a solution to (2) for every x(0)${\displaystyle \mathbb {R} ^{n}}$ if and only if T is semi-convergent. In this case, the splitting (5) is called a semi-convergent splitting of A.[15]

## References

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