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

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

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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

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

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into an equivalent system of the form

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

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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

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

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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

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

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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

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

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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