מתוך ויקיספר, אוסף הספרים והמדריכים החופשי
שיחלוף (באנגלית: transpose) של מטריצה כלשהי
A
∈
F
m
×
n
{\displaystyle A\in \mathbb {F} ^{m\times n}}
A
t
{\displaystyle A^{t}}
A
T
{\displaystyle A^{T}}
A
t
∈
F
n
×
m
{\displaystyle A^{t}\in \mathbb {F} ^{n\times m}}
[
A
t
]
i
,
j
=
[
A
]
j
,
i
{\displaystyle [A^{t}]_{i,j}=[A]_{j,i}}
(
1
2
3
4
5
6
)
t
=
(
1
4
2
5
3
6
)
{\displaystyle {\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}}^{t}={\begin{pmatrix}1&4\\2&5\\3&6\end{pmatrix}}}
הגדרה 4: המטריצה המשוחלפת
כאשר
A
∈
M
m
,
n
(
F
)
{\displaystyle A\in M_{m,n}(\mathbb {F} )}
A
T
{\displaystyle A^{T}}
a
i
,
j
{\displaystyle a_{i,j}}
a
j
,
i
{\displaystyle a_{j,i}}
A
{\displaystyle A}
m
×
n
{\displaystyle m\times n}
A
T
{\displaystyle A^{T}}
n
×
m
{\displaystyle n\times m}
דוגמא: ניקח את המטריצה
A
=
(
3
4
5
6
7
1
)
{\displaystyle A={\begin{pmatrix}3&4&5\\6&7&1\end{pmatrix}}}
A
T
=
(
3
6
4
7
5
1
)
{\displaystyle A^{T}={\begin{pmatrix}3&6\\4&7\\5&1\end{pmatrix}}}
[ עריכה ] יהי
S
=
{
(
1
2
3
)
}
{\displaystyle S=\left\{{\begin{pmatrix}1\\2\\3\end{pmatrix}}\right\}}
V
=
R
3
{\displaystyle V=\mathbb {R} ^{3}}
(
1
|
1
0
0
2
|
0
1
0
3
|
0
0
1
)
→
(
1
|
1
0
0
0
|
−
2
1
0
0
|
−
3
0
1
)
{\displaystyle {\begin{pmatrix}1&|&1&0&0\\2&|&0&1&0\\3&|&0&0&1\end{pmatrix}}\to {\begin{pmatrix}1&|&1&0&0\\0&|&-2&1&0\\0&|&-3&0&1\end{pmatrix}}}
s
p
a
n
(
(
1
2
3
)
)
=
{
(
x
1
x
2
x
3
)
∈
R
3
∣
{
−
2
x
1
+
x
2
=
0
−
3
x
1
+
x
3
=
0
}
{\displaystyle span\left({\begin{pmatrix}1\\2\\3\end{pmatrix}}\right)=\left\{{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\in \mathbb {R} ^{3}\mid {\begin{cases}-2x_{1}+x_{2}=0\\-3x_{1}+x_{3}=0\end{cases}}\right\}}
נמצא קבוצה פורשת של מטריצה משוחלפת:
נכפיל
(
1
2
3
)
(
x
1
x
2
x
3
)
=
0
⇒
x
1
+
2
x
2
+
3
x
3
=
0
{\displaystyle {\begin{pmatrix}1&2&3\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}=0\Rightarrow x_{1}+2x_{2}+3x_{3}=0}
לאחר העברת אגפים
x
1
=
−
2
x
2
−
3
x
3
{\displaystyle x_{1}=-2x_{2}-3x_{3}}
{
(
−
2
t
1
−
3
t
2
t
1
t
2
)
∣
t
1
,
t
2
∈
R
}
=
s
p
a
n
(
(
−
2
1
0
)
,
(
−
3
0
1
)
)
.
{\displaystyle \left\{{\begin{pmatrix}-2t_{1}-3t_{2}\\t_{1}\\t_{2}\end{pmatrix}}\mid t_{1},t_{2}\in \mathbb {R} \right\}=span\left({\begin{pmatrix}-2\\1\\0\end{pmatrix}},{\begin{pmatrix}-3\\0\\1\end{pmatrix}}\right).}
נגדיר
l
1
(
x
1
x
2
x
3
)
=
−
2
x
1
+
2
{\displaystyle l_{1}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}=-2x_{1}+2}
l
2
(
x
1
x
2
x
3
)
=
−
3
x
1
+
x
3
{\displaystyle l_{2}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}=-3x_{1}+x_{3}}
(
l
1
,
l
2
)
{\displaystyle \left(l_{1},l_{2}\right)}
{
(
x
1
x
2
x
3
)
}
0
{\displaystyle \left\{{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\right\}^{0}}
s
p
a
n
(
(
1
2
3
)
)
=
{
x
∈
R
3
∣
{
l
1
(
x
)
=
0
l
2
(
x
)
=
0
}
.
{\displaystyle span\left({\begin{pmatrix}1\\2\\3\end{pmatrix}}\right)=\left\{x\in \mathbb {R} ^{3}\mid {\begin{cases}l_{1}\left(x\right)=0\\l_{2}\left(x\right)=0\end{cases}}\right\}.}
(
A
t
)
t
=
A
{\displaystyle (A^{t})^{t}=A}
(
A
+
B
)
t
=
A
t
+
B
t
{\displaystyle (A+B)^{t}=A^{t}+B^{t}}
(
α
A
)
t
=
α
(
A
t
)
{\displaystyle (\alpha A)^{t}=\alpha (A^{t})}
tr
(
A
t
)
=
tr
(
A
)
{\displaystyle {\mbox{tr}}(A^{t})={\mbox{tr}}(A)}
(
A
B
)
t
=
B
t
A
t
{\displaystyle (AB)^{t}=B^{t}A^{t}}
משפט 2: אם המכפלה
A
B
{\displaystyle AB}
(
A
B
)
T
=
B
T
A
T
{\displaystyle (AB)^{T}=B^{T}A^{T}}
הוכחה:
[
(
A
B
)
T
]
i
j
=
[
A
B
]
j
i
=
∑
l
=
1
n
a
j
l
b
l
i
=
∑
l
=
1
n
b
l
i
a
j
l
=
∑
i
=
1
n
[
B
T
]
i
l
[
A
T
]
l
j
=
[
B
T
A
T
]
i
j
{\displaystyle \left[\left(AB\right)^{T}\right]_{ij}=\left[AB\right]_{ji}=\sum _{l=1}^{n}\ a_{jl}b_{li}=\sum \ _{l=1}^{n}\ b_{li}a_{jl}=\sum _{i=1}^{n}\ \left[B^{T}\right]_{il}\left[A^{T}\right]_{lj}=\left[B^{T}A^{T}\right]_{ij}}
![{\displaystyle [A^{t}]_{i,j}=[A]_{j,i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04d08986b81e31f4a636c57eb58039693cd5f564)
![{\displaystyle \left[\left(AB\right)^{T}\right]_{ij}=\left[AB\right]_{ji}=\sum _{l=1}^{n}\ a_{jl}b_{li}=\sum \ _{l=1}^{n}\ b_{li}a_{jl}=\sum _{i=1}^{n}\ \left[B^{T}\right]_{il}\left[A^{T}\right]_{lj}=\left[B^{T}A^{T}\right]_{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/495041e12e9e45c150cffa74daa832da5009a070)
