Convergent matrix

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In linear algebra, a convergent matrix is a matrix that converges to the zero matrix under matrix exponentiation.

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.

We call an n × n matrix T a convergent matrix if

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

Let

${\displaystyle {\begin{aligned}&\mathbf {T} ={\begin{pmatrix}{\frac {1}{4}}&{\frac {1}{2}}\\[4pt]0&{\frac {1}{4}}\end{pmatrix}}.\end{aligned}}}${\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} ^{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}}}

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

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

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

and

${\displaystyle \lim _{k\to \infty }{\frac {k}{2^{2k-1}}}=0,}${\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.

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

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

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

into an equivalent system of the form

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

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

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

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]}

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

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⁾ ∈ ${\displaystyle \mathbb {R} ^{n}}${\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]

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

• 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. Bibcode:1977SJNA...14..699M. 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, Bibcode:1962mia..book.....V, LCCN 62-21277 .

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