Vide etiam paginam discretivam:
matrix .
Matrix in mathematice est tabula numerorum formae rectangularis. Hi numeri matricis elementa vel componentia scilicet elementa vel fortasse relata scilicet elementa in matrice vocantur.
Matricis nomen mathematicus Anglicus Iacobus Ioseph Sylvester finxit.
Typus matricis definitur numero ordinum transversorum vel lineolarum et numero ut ita dicam columnarum vel ordinum directorum. Quodcumque matricis elementum ad quemdam ordinem tranversum et ad quemdam directum pertinet. Haec matrix exempli gratia habet ordines duos transversos et tres directos et designatur nomine matrix
2
×
3
{\displaystyle 2\times 3}
(
a
11
a
12
a
13
a
21
a
22
a
23
)
{\displaystyle {\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\end{pmatrix}}}
[
a
11
a
12
a
13
a
21
a
22
a
23
]
{\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\end{bmatrix}}}
Si ordinum numerus directorum est idem quam transversorum matrix vocatur quadrata.
His cum matricibus operationes mathematicae perfieri possunt sicut additio , subtractio vel multiplicatio . Alia operatio specialis quae cum illis matricibus fieri potest, quae sunt regulares, est inversio . Et alia operatio est transpositio .
Si duae matrices sunt eiusdem typi alia alii addi possunt vel alia matrix ab alia subrahi potest.
Haec est summa duarum matricum:
m
×
n
{\displaystyle m\times n}
A
+
B
:=
(
a
i
j
+
b
i
j
)
i
=
1
,
…
,
m
;
j
=
1
,
…
,
n
{\displaystyle A+B:=(a_{ij}+b_{ij})_{i=1,\ldots ,m;\ j=1,\ldots ,n}}
Ad exemplum:
(
1
−
3
2
1
2
7
)
+
(
0
3
5
2
1
−
1
)
=
(
1
+
0
−
3
+
3
2
+
5
1
+
2
2
+
1
7
+
(
−
1
)
)
=
(
1
0
7
3
3
6
)
{\displaystyle {\begin{pmatrix}1&-3&2\\1&2&7\end{pmatrix}}+{\begin{pmatrix}0&3&5\\2&1&-1\end{pmatrix}}={\begin{pmatrix}1+0&-3+3&2+5\\1+2&2+1&7+(-1)\end{pmatrix}}={\begin{pmatrix}1&0&7\\3&3&6\end{pmatrix}}}
Matrix cum numero scalari multiplicatur eo modo, quod omnia matricis elementa multiplicantur cum numero scalari:
λ
⋅
A
:=
(
λ
⋅
a
i
j
)
i
=
1
,
…
,
m
;
j
=
1
,
…
,
n
{\displaystyle \lambda \cdot A:=(\lambda \cdot a_{ij})_{i=1,\ldots ,m;\ j=1,\ldots ,n}}
Hoc in exemplo numerus scalaris est 5:
5
⋅
(
1
−
3
2
1
2
7
)
=
(
5
⋅
1
5
⋅
(
−
3
)
5
⋅
2
5
⋅
1
5
⋅
2
5
⋅
7
)
=
(
5
−
15
10
5
10
35
)
{\displaystyle 5\cdot {\begin{pmatrix}1&-3&2\\1&2&7\end{pmatrix}}={\begin{pmatrix}5\cdot 1&5\cdot (-3)&5\cdot 2\\5\cdot 1&5\cdot 2&5\cdot 7\end{pmatrix}}={\begin{pmatrix}5&-15&10\\5&10&35\end{pmatrix}}}
Productum matricum A et B: elementum flavum colore producti est
p
12
=
a
11
b
12
+
a
12
b
21
{\displaystyle p_{12}=a_{11}b_{12}+a_{12}b_{21}}
p
33
=
a
31
b
13
+
a
32
b
23
{\displaystyle p_{33}=a_{31}b_{13}+a_{32}b_{23}}
Duae matrices multiplicari possunt si numerus ordinum transversorum laevae matricis est idem quam numerus ordinum directorum dextrae.
Productum matricis
l
×
m
{\displaystyle l\times m}
A
=
(
a
i
j
)
i
=
1
…
l
,
j
=
1
…
m
{\displaystyle A=(a_{ij})_{i=1\ldots l,\;j=1\ldots m}}
m
×
n
{\displaystyle m\times n}
B
=
(
b
i
j
)
i
=
1
…
m
,
j
=
1
…
n
{\displaystyle B=(b_{ij})_{i=1\ldots m,\;j=1\ldots n}}
l
×
n
{\displaystyle l\times n}
C
=
(
c
i
j
)
i
=
1
…
l
,
j
=
1
…
n
{\displaystyle C=(c_{ij})_{i=1\ldots l,\;j=1\ldots n}}
vectore directo aliae matricis:
c
i
j
=
∑
k
=
1
m
a
i
k
⋅
b
k
j
{\displaystyle c_{ij}=\sum _{k=1}^{m}a_{ik}\cdot b_{kj}}
Computationis exemplum:
(
1
2
3
4
5
6
)
⋅
(
6
−
1
3
2
0
−
3
)
=
(
1
⋅
6
+
2
⋅
3
+
3
⋅
0
1
⋅
(
−
1
)
+
2
⋅
2
+
3
⋅
(
−
3
)
4
⋅
6
+
5
⋅
3
+
6
⋅
0
4
⋅
(
−
1
)
+
5
⋅
2
+
6
⋅
(
−
3
)
)
=
(
12
−
6
39
−
12
)
{\displaystyle {\begin{pmatrix}1&2&3\\4&5&6\\\end{pmatrix}}\cdot {\begin{pmatrix}6&-1\\3&2\\0&-3\end{pmatrix}}={\begin{pmatrix}1\cdot 6+2\cdot 3+3\cdot 0&1\cdot (-1)+2\cdot 2+3\cdot (-3)\\4\cdot 6+5\cdot 3+6\cdot 0&4\cdot (-1)+5\cdot 2+6\cdot (-3)\\\end{pmatrix}}={\begin{pmatrix}12&-6\\39&-12\end{pmatrix}}}
Matrix quadratica A sit:
A
=
(
1
2
3
4
5
6
7
8
9
)
{\displaystyle A={\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}}}
Matrix A cum se ipsa multiplicatur sic:
(
1
2
3
4
5
6
7
8
9
)
⋅
(
1
2
3
4
5
6
7
8
9
)
=
(
1
⋅
1
+
2
⋅
4
+
3
⋅
7
1
⋅
2
+
2
⋅
5
+
3
⋅
8
1
⋅
3
+
2
⋅
6
+
3
⋅
9
4
⋅
1
+
5
⋅
4
+
6
⋅
7
4
⋅
2
+
5
⋅
5
+
6
⋅
8
4
⋅
3
+
5
⋅
6
+
6
⋅
9
7
⋅
1
+
8
⋅
4
+
9
⋅
7
7
⋅
2
+
8
⋅
5
+
9
⋅
8
7
⋅
3
+
8
⋅
6
+
9
⋅
9
)
=
(
30
36
42
66
81
96
102
122
150
)
{\displaystyle {\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}}\cdot {\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}}={\begin{pmatrix}1\cdot 1+2\cdot 4+3\cdot 7&1\cdot 2+2\cdot 5+3\cdot 8&1\cdot 3+2\cdot 6+3\cdot 9\\4\cdot 1+5\cdot 4+6\cdot 7&4\cdot 2+5\cdot 5+6\cdot 8&4\cdot 3+5\cdot 6+6\cdot 9\\7\cdot 1+8\cdot 4+9\cdot 7&7\cdot 2+8\cdot 5+9\cdot 8&7\cdot 3+8\cdot 6+9\cdot 9\\\end{pmatrix}}={\begin{pmatrix}30&36&42\\66&81&96\\102&122&150\end{pmatrix}}}
Eo modo matrix A est elevata ad potentiam secundam.
Matrix idemfactor est matrix quadrata diagonalis , cuius omnia elementa in diagonale principali sunt 1 et omnia alia sunt 0; signum talis matricis est I, vel In ubi n est numerus ordinum et columnum.
I
3
=
(
1
0
0
0
1
0
0
0
1
)
{\displaystyle I_{3}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\\\end{pmatrix}}}
Deinde, omni matrice A, AI = IA = A.
Si est matrix
A
=
(
a
i
j
)
{\displaystyle A=\left(a_{ij}\right)}
m
×
n
{\displaystyle m\times n}
A
T
=
(
a
j
i
)
{\displaystyle A^{T}=\left(a_{ji}\right)}
n
×
m
{\displaystyle n\times m}
A
=
(
a
11
…
a
1
n
⋮
⋱
⋮
a
m
1
…
a
m
n
)
{\displaystyle A={\begin{pmatrix}a_{11}&\dots &a_{1n}\\\vdots &\ddots &\vdots \\a_{m1}&\dots &a_{mn}\end{pmatrix}}}
habet matricem transformatam
A
T
=
(
a
11
…
a
m
1
⋮
⋱
⋮
a
1
n
…
a
m
n
)
.
{\displaystyle A^{T}={\begin{pmatrix}a_{11}&\dots &a_{m1}\\\vdots &\ddots &\vdots \\a_{1n}&\dots &a_{mn}\end{pmatrix}}.}
Exemplum:
(
1
8
−
3
4
−
2
5
)
T
=
(
1
4
8
−
2
−
3
5
)
.
{\displaystyle {\begin{pmatrix}1&8&-3\\4&-2&5\end{pmatrix}}^{T}={\begin{pmatrix}1&4\\8&-2\\-3&5\end{pmatrix}}.}
Nexus interni
