# 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

$\lim_{k \to \infty}( \bold T^k)_{ij} = \bold 0, \quad (1)$

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

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## Example

Let

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

Computing successive powers of T, we obtain

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

and, in general,

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

Since

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

and

$\lim_{k \to \infty} \frac{k}{2^{2k - 1}} = 0,$

T is a convergent matrix. Note that ρ(T) = 14, where ρ(T) represents the spectral radius of T, since 14 is the only eigenvalue of T.

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## Characterizations

Let T be an n × n matrix. The following properties are equivalent to T being a convergent matrix:

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

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

$\bold{Ax} = \bold{b} \quad (2)$

into an equivalent system of the form

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

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

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

for each k ≥ 0.[8][9] For any initial vector x(0)$\mathbb{R}^n$, the sequence $\lbrace \bold{x}^{ \left( k \right) } \rbrace _{k = 0}^{\infty}$ 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]

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## 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 (2) above, with A non-singular, the matrix A can be split, i.e., written as a difference

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

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]

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## Semi-convergent matrix

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

$\lim_{k \to \infty} \bold T^k \quad (6)$

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)$\mathbb{R}^n$ if and only if T is semi-convergent. In this case, the splitting (5) is called a semi-convergent splitting of A.[15]

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## 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)
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## 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.
• Varga, Richard S. (1960). "Factorization and Normalized Iterative Methods". In Langer, Rudolph E. Boundary Problems in Differential Equations. Madison: University of Wisconsin Press. pp. 121–142. LCCN 60-60003.
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