Convergent matrix

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

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. (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 .