Matrize norma bat, bektoreena bezala,
‖
⋅
‖
{\displaystyle \lVert \cdot \rVert }
‖
A
‖
>
0
,
∀
A
≠
0.
{\displaystyle \lVert A\rVert >0,\forall A\neq 0.}
‖
c
A
‖
=
|
c
|
⋅
‖
A
‖
,
∀
c
∈
R
.
{\displaystyle \lVert cA\rVert =\left\vert c\right\vert \cdot \lVert A\rVert ,\forall c\in {\displaystyle \mathbb {R} }.}
‖
A
+
B
‖
≤
‖
A
‖
+
‖
B
‖
.
{\displaystyle \lVert A+B\rVert \leq \lVert A\rVert +\lVert B\rVert .}
A eta B
K
m
×
n
{\displaystyle K^{m\times n}}
Gainera, matrizea karratua den kasuetan; hau da, m=n hurrengo propietatea betetzen dela esan dezakegu:
‖
A
B
‖
≤
‖
A
‖
⋅
‖
B
‖
.
{\displaystyle \lVert AB\rVert \leq \lVert A\rVert \cdot \lVert B\rVert .}
Izan bitez A matrize bat eta
‖
⋅
‖
{\displaystyle \lVert \cdot \rVert }
‖
A
‖
{\displaystyle \lVert A\rVert }
‖
A
‖
=
max
x
≠
0
‖
A
x
‖
x
{\displaystyle \lVert A\rVert =\max \limits _{x\neq 0}{\frac {\lVert Ax\rVert }{x}}}
Jarraian bektoreen bat-, bi- eta infinitu-normek eragindako matrize normak adieraziko ditugu:
‖
A
‖
1
=
max
j
‖
A
:
,
j
‖
1
{\displaystyle \lVert A\rVert _{1}=\max \limits _{j}\lVert A_{:,j}\rVert _{1}}
‖
A
‖
2
=
σ
1
(
A
)
{\displaystyle \lVert A\rVert _{2}=\sigma _{1}(A)}
‖
A
‖
∞
=
max
i
‖
A
i
,
:
‖
1
{\displaystyle \lVert A\rVert _{\infty }=\max \limits _{i}\lVert A_{i,:}\rVert _{1}}
A simetrikoa den kasuetan,
‖
A
‖
2
=
max
1
≤
i
≤
n
|
λ
i
|
{\displaystyle \lVert A\rVert _{2}=\max \limits _{1\leq i\leq n}\left\vert \lambda _{i}\right\vert }
λ
i
,
1
≤
i
≤
n
{\displaystyle \lambda _{i},1\leq i\leq n}
‖
⋅
‖
b
{\displaystyle \lVert \cdot \rVert _{b}}
‖
⋅
‖
m
{\displaystyle \lVert \cdot \rVert _{m}}
‖
A
x
‖
b
≤
‖
A
‖
m
‖
x
‖
b
.
{\displaystyle \lVert Ax\rVert _{b}\leq \lVert A\rVert _{m}\lVert x\rVert _{b}.}
Bektore-norma eta berak eragindako matrize-norma beti izango dira bateragarriak, baina ez eragindako matrize norma bat ere badago, Frobenius en norma:
‖
A
‖
F
=
(
∑
i
=
1
m
∑
j
=
1
n
a
i
,
j
2
)
1
2
{\displaystyle \lVert A\rVert _{F}=(\sum _{i=1}^{m}\sum _{j=1}^{n}a_{i,j}^{2})^{\frac {1}{2}}}
Norma hau bateragarria izanik bektore-norma euklidearrarekin:
‖
A
x
‖
2
≤
‖
A
‖
F
‖
x
‖
2
.
{\displaystyle \lVert Ax\rVert _{2}\leq \lVert A\rVert _{F}\lVert x\rVert _{2}.}
Edozein
A
∈
R
m
×
n
{\displaystyle A\in {\displaystyle \mathbb {R} }^{m\times n}}
‖
A
‖
2
≤
‖
A
‖
F
≤
n
‖
A
‖
2
.
{\displaystyle \lVert A\rVert _{2}\leq \lVert A\rVert _{F}\leq {\sqrt {n}}\lVert A\rVert _{2}.}
1
n
‖
A
‖
∞
≤
‖
A
‖
2
≤
m
‖
A
‖
∞
.
{\displaystyle {\frac {1}{\sqrt {n}}}\lVert A\rVert _{\infty }\leq \lVert A\rVert _{2}\leq {\sqrt {m}}\lVert A\rVert _{\infty }.}
1
m
‖
A
‖
1
≤
‖
A
‖
2
≤
n
‖
A
‖
1
.
{\displaystyle {\frac {1}{\sqrt {m}}}\lVert A\rVert _{1}\leq \lVert A\rVert _{2}\leq {\sqrt {n}}\lVert A\rVert _{1}.}
Propietate honen arabera, bektore edo matrize norma ezberdinek balio ezberdinak izan ditzaketen arren, baliokidetzat har daitezke, baten balio ezagutuz beste norma batena borna baitezakegu.
