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
(
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)
