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

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

(1)

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

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) = 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. $\lim_{k \to \infty} \| \bold T^k \| = 0,$ 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]

Iterative methods

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⁾ ∈ $\mathbb {R} ^{n}$, 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, that is, T is a convergent matrix.^[10][11]

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, 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⁻¹ ≥ 0 and C ≥ 0, that is, B⁻¹ and C have only nonnegative entries. If the splitting (5) is a regular splitting of the matrix A and A⁻¹ ≥ 0, 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⁾ ∈ $\mathbb {R} ^{n}$ if and only if T is semi-convergent. In this case, the splitting (5) is called a semi-convergent splitting of A.^[15]

See also

• Gauss–Seidel method
• Jacobi method
• List of matrices
• Nilpotent matrix
• Successive over-relaxation

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. 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.