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 × n matrix T a convergent matrix if

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

(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) = 1/4, where ρ(T) represents the spectral radius of T, since 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:

1. ${\displaystyle \lim _{k\to \infty }\|\mathbf {T} ^{k}\|=0,}$ for some natural norm;
2. ${\displaystyle \lim _{k\to \infty }\|\mathbf {T} ^{k}\|=0,}$ for all natural norms;
3. ${\displaystyle \rho (\mathbf {T} )<1}$;
4. ${\displaystyle \lim _{k\to \infty }\mathbf {T} ^{k}\mathbf {x} =\mathbf {0} ,}$ for every x.[4][5][6][7]

## Iterative methods

 Main article: Iterative method

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

${\displaystyle \mathbf {Ax} =\mathbf {b} }$

(2)

into an equivalent system of the form

${\displaystyle \mathbf {x} =\mathbf {Tx} +\mathbf {c} }$

(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

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

(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, that is, T is a convergent matrix.[10][11]

### Regular splitting

 Main article: Matrix 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 (2) above, with A non-singular, the matrix A can be split, that is, written as a difference

${\displaystyle \mathbf {A} =\mathbf {B} -\mathbf {C} }$

(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, that is, 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

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

(6)

exists.[14] If A is possibly singular but (2) is consistent, that is, 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]

## Notes

1. ^ Burden & Faires (1993, p. 404)
2. ^ Isaacson & Keller (1994, p. 14)
3. ^ Varga (1962, p. 13)
4. ^ Burden & Faires (1993, p. 404)
5. ^ Isaacson & Keller (1994, pp. 14, 63)
6. ^ Varga (1960, p. 122)
7. ^ Varga (1962, p. 13)
8. ^ Burden & Faires (1993, p. 406)
9. ^ Varga (1962, p. 61)
10. ^ Burden & Faires (1993, p. 412)
11. ^ Isaacson & Keller (1994, pp. 62–63)
12. ^ Varga (1960, pp. 122–123)
13. ^ Varga (1962, p. 89)
14. ^ Meyer & Plemmons (1977, p. 699)
15. ^ Meyer & Plemmons (1977, p. 700)

## References

• Burden, Richard L.; Faires, J. Douglas (1993), Numerical Analysis (5th ed.), Boston: Prindle, Weber and Schmidt, ISBN 0-534-93219-3.
• Isaacson, Eugene; Keller, Herbert Bishop (1994), Analysis of Numerical Methods, New York: Dover, ISBN 0-486-68029-0.
• Carl D. Meyer, Jr.; R. J. Plemmons (Sep 1977). "Convergent Powers of a Matrix with Applications to Iterative Methods for Singular Linear Systems". SIAM Journal on Numerical Analysis. 14 (4): 699–705. doi:10.1137/0714047.
• Varga, Richard S. (1960). "Factorization and Normalized Iterative Methods". In Langer, Rudolph E. (ed.). Boundary Problems in Differential Equations. Madison: University of Wisconsin Press. pp. 121–142. LCCN 60-60003.
• Varga, Richard S. (1962), Matrix Iterative Analysis, New Jersey: Prentice–Hall, LCCN 62-21277.