线性代数 中，收敛矩阵 是在求幂过程中收敛到零矩阵 的矩阵。
矩阵 T 的幂随次数增加而变小时（即T 的所有项都趋近于0），T 收敛到零矩阵。可逆矩阵 A 的正则分裂 会产生收敛矩阵T 。A 的半收敛分裂会产生半收敛矩阵T 。将T 用于一般的迭代法 ，则对任意初向量都是收敛的；半收敛的T 则要初向量满足特定条件才收敛。
n 阶方阵T 若满足
∀
i
=
1
,
2
,
…
,
n
,
j
=
1
,
2
,
…
,
n
,
lim
k
→
∞
(
T
k
)
i
j
=
0
,
{\displaystyle \forall i=1,\ 2,\ \dots ,\ n,\ j=1,\ 2,\ \dots ,\ n,\ \lim _{k\to \infty }(\mathbf {T} ^{k})_{ij}=0,}
1
则称T 是是收敛矩阵 。[ 1] [ 2] [ 3]
令
T
=
(
1
4
1
2
0
1
4
)
.
{\displaystyle {\begin{aligned}&\mathbf {T} ={\begin{pmatrix}{\frac {1}{4}}&{\frac {1}{2}}\\[4pt]0&{\frac {1}{4}}\end{pmatrix}}.\end{aligned}}}
T 的幂是
T
2
=
(
1
16
1
4
0
1
16
)
,
T
3
=
(
1
64
3
32
0
1
64
)
,
T
4
=
(
1
256
1
32
0
1
256
)
,
T
5
=
(
1
1024
5
512
0
1
1024
)
,
{\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}}}
T
6
=
(
1
4096
3
1024
0
1
4096
)
,
{\displaystyle {\begin{aligned}\mathbf {T} ^{6}={\begin{pmatrix}{\frac {1}{4096}}&{\frac {3}{1024}}\\[4pt]0&{\frac {1}{4096}}\end{pmatrix}},\end{aligned}}}
综之，
T
k
=
(
(
1
4
)
k
k
2
2
k
−
1
0
(
1
4
)
k
)
.
{\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}}}
由于
lim
k
→
∞
(
1
4
)
k
=
0
{\displaystyle \lim _{k\to \infty }\left({\frac {1}{4}}\right)^{k}=0}
lim
k
→
∞
k
2
2
k
−
1
=
0
,
{\displaystyle \lim _{k\to \infty }{\frac {k}{2^{2k-1}}}=0,}
T 是收敛矩阵。注意其谱半径
ρ
(
T
)
=
1
4
{\displaystyle \rho (\mathbf {T} )={\frac {1}{4}}}
1
4
{\displaystyle {\frac {1}{4}}}
T 唯一的特征值 。
设T 是n 阶方阵，则下列表述等价于T 的收敛矩阵：
对某自然范数，
lim
k
→
∞
‖
T
k
‖
=
0
{\displaystyle \lim _{k\to \infty }\|\mathbf {T} ^{k}\|=0}
对所有自然范数，
lim
k
→
∞
‖
T
k
‖
=
0
{\displaystyle \lim _{k\to \infty }\|\mathbf {T} ^{k}\|=0}
ρ
(
T
)
<
1
{\displaystyle \rho (\mathbf {T} )<1}
∀
x
,
lim
k
→
∞
T
k
x
=
0
{\displaystyle \forall \mathbb {x} ,\ \lim _{k\to \infty }\mathbf {T} ^{k}\mathbf {x} =\mathbf {0} }
[ 4] [ 5] [ 6] [ 7]
一般的迭代法 包含将线性方程组
A
x
=
b
{\displaystyle \mathbf {Ax} =\mathbf {b} }
2
转为等价方程组
x
=
T
x
+
c
{\displaystyle \mathbf {x} =\mathbf {Tx} +\mathbf {c} }
3
的过程。选定初向量
x
(
0
)
{\displaystyle \mathbf {x} ^{(0)}}
∀
k
≥
0
,
x
(
k
+
1
)
=
T
x
(
k
)
+
c
{\displaystyle \forall k\geq 0,\ \mathbf {x} ^{(k+1)}=\mathbf {Tx} ^{(k)}+\mathbf {c} }
4
[ 8] [ 9]
对任意初向量
x
(
0
)
∈
R
n
{\displaystyle \mathbf {x} ^{(0)}\in \mathbb {R} ^{n}}
{
x
(
k
)
}
k
=
0
∞
{\displaystyle \lbrace \mathbf {x} ^{\left(k\right)}\rbrace _{k=0}^{\infty }}
4
∀
k
≥
0
,
c
≠
0
{\displaystyle \forall k\geq 0,\ \mathbf {c} \neq 0}
ρ
(
T
)
<
1
{\displaystyle \rho (\mathbf {T} )<1}
3 T 是收敛矩阵。[ 10] [ 11]
矩阵分裂 是用多个矩阵的和或差表示矩阵。对(2 A 可逆，则A 就可分裂为
A
=
B
−
C
{\displaystyle \mathbf {A} =\mathbf {B} -\mathbf {C} }
5
于是(2 4
B
−
1
≥
0
,
C
≥
0
{\displaystyle \mathbf {B} ^{-1}\geq \mathbf {0} ,\ \mathbf {C} \geq \mathbf {0} }
5 A的正则分裂 ；即
B
−
1
,
C
{\displaystyle \mathbf {B} ^{-1},\ \mathbf {C} }
5 A 的正则分裂、且
A
−
1
≥
0
{\displaystyle \mathbf {A} ^{-1}\geq \mathbf {0} }
ρ
(
T
)
<
1
{\displaystyle \rho (\mathbf {T} )<1}
T 是收敛矩阵，迭代法(4 [ 12] [ 13]
n 阶方阵T ，若极限
lim
k
→
∞
T
k
{\displaystyle \lim _{k\to \infty }\mathbf {T} ^{k}}
6
存在，则称之为半收敛矩阵 。[ 14] A 可能奇异，而(2 b 在A 的范围内，则当且仅当T 是半收敛矩阵时，对任何初向量
x
(
0
)
∈
R
n
{\displaystyle \mathbf {x} ^{(0)}\in \mathbb {R} ^{n}}
4 2 5 A 的半收敛分裂 。[ 15]
Burden, Richard L.; Faires, J. Douglas, Numerical Analysis Prindle, Weber and Schmidt , 1993, ISBN 0-534-93219-3 Isaacson, Eugene; Keller, Herbert Bishop, Analysis of Numerical Methods, New York: Dover , 1994, ISBN 0-486-68029-0 Carl D. Meyer, Jr.; R. J. Plemmons. Convergent Powers of a Matrix with Applications to Iterative Methods for Singular Linear Systems. SIAM Journal on Numerical Analysis . Sep 1977, 14 (4): 699–705. doi:10.1137/0714047 Varga, Richard S. Factorization and Normalized Iterative Methods. Langer, Rudolph E. (编). Boundary Problems in Differential Equations. Madison: University of Wisconsin Press . 1960: 121–142. LCCN 60-60003 Varga, Richard S., Matrix Iterative Analysis, New Jersey: Prentice–Hall , 1962, LCCN 62-21277 
![{\displaystyle {\begin{aligned}&\mathbf {T} ={\begin{pmatrix}{\frac {1}{4}}&{\frac {1}{2}}\\[4pt]0&{\frac {1}{4}}\end{pmatrix}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fb9c103c5b4a081500fba565c48a4ec23fa575d)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5128aa46f45514aec3c08b06092d9429cf123b6)
![{\displaystyle {\begin{aligned}\mathbf {T} ^{6}={\begin{pmatrix}{\frac {1}{4096}}&{\frac {3}{1024}}\\[4pt]0&{\frac {1}{4096}}\end{pmatrix}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26be82305d5cb1457c42a7fb721ca9208ac6857c)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af62f4a4772caeb09656406c54305314ee2ad3d2)
5th, Boston: Prindle, Weber and Schmidt, 1993, ISBN 0-534-93219-3.