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Polinomska matrika (tudi matrika mnogočlenikov ali polinomov) je matrika , ki ima za elemente polinome z eno (univariantna) ali več (multivariantna) spremenljivkami . Posebno obliko imenujemo tudi matrika λ . To je matrika, katere elementi so polinomi spremenljivke
λ
{\displaystyle \lambda \,}
λ
{\displaystyle \lambda \,}
Univariantna polinomska matrika stopnje
p
{\displaystyle p\,}
P
=
∑
n
=
0
p
A
(
n
)
x
n
=
A
(
0
)
+
A
(
1
)
x
+
A
(
2
)
x
2
+
⋯
+
A
(
p
)
x
p
{\displaystyle P=\sum _{n=0}^{p}A(n)x^{n}=A(0)+A(1)x+A(2)x^{2}+\cdots +A(p)x^{p}}
kjer je
A
(
i
)
{\displaystyle A(i)\,}
A
(
p
)
{\displaystyle A(p)\,}
Primer matrike λ je
A
(
λ
)
=
[
a
11
(
λ
)
a
12
(
λ
)
⋯
a
1
n
(
λ
)
a
21
(
λ
)
a
22
(
λ
)
⋯
a
2
n
(
λ
)
⋮
⋮
⋱
⋮
a
n
1
(
λ
)
a
n
2
(
λ
)
⋯
a
n
n
(
λ
)
]
,
a
i
j
(
λ
)
=
a
i
j
(
l
)
λ
l
+
a
i
j
(
l
−
1
)
λ
l
−
1
+
⋯
+
a
i
j
(
1
)
λ
+
a
i
j
(
0
)
.
{\displaystyle A\left(\lambda \right)={\begin{bmatrix}a_{11}(\lambda )&a_{12}(\lambda )&\cdots &a_{1n}(\lambda )\\a_{21}(\lambda )&a_{22}(\lambda )&\cdots &a_{2n}(\lambda )\\\vdots &\vdots &\ddots &\vdots \\a_{n1}(\lambda )&a_{n2}(\lambda )&\cdots &a_{nn}(\lambda )\end{bmatrix}},\quad a_{ij}(\lambda )=a_{ij}^{(l)}\lambda ^{l}+a_{ij}^{(l-1)}\lambda ^{l-1}+\cdots +a_{ij}^{(1)}\lambda +a_{ij}^{(0)}.}
kjer je
l
{\displaystyle l\,}
a
i
j
{\displaystyle a_{ij}\,}
Primer takšne matrike je
A
(
λ
)
=
[
λ
4
+
λ
2
+
λ
−
1
λ
3
+
λ
2
+
λ
+
2
2
λ
3
−
λ
2
λ
2
+
2
λ
]
=
[
1
0
0
0
]
λ
4
+
[
0
1
2
0
]
λ
3
+
[
1
1
0
2
]
λ
2
+
[
1
1
−
1
2
]
λ
+
[
−
1
2
0
0
]
.
{\displaystyle A\left(\lambda \right)={\begin{bmatrix}\lambda ^{4}+\lambda ^{2}+\lambda -1&\lambda ^{3}+\lambda ^{2}+\lambda +2\\2\lambda ^{3}-\lambda &2\lambda ^{2}+2\lambda \end{bmatrix}}={\begin{bmatrix}1&0\\0&0\end{bmatrix}}\lambda ^{4}+{\begin{bmatrix}0&1\\2&0\end{bmatrix}}\lambda ^{3}+{\begin{bmatrix}1&1\\0&2\end{bmatrix}}\lambda ^{2}+{\begin{bmatrix}1&1\\-1&2\end{bmatrix}}\lambda +{\begin{bmatrix}-1&2\\0&0\end{bmatrix}}.}
