What's a simple proof for $^{t}(A \cdot B) = ^{t}B \cdot ^{t}A$ with $A \in M_{m,n}, B \in M_{n, p}$ ?
Proof for $^{t}(A \cdot B) = ^{t}B \cdot ^{t}A$?
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1Is that transposition? – 2011-12-30
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0And is the $\cdot$ matrix multiplication? – 2011-12-30
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0Yes to both your questions – 2011-12-30
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0Matrix multiplication is usually not written with a $\cdot$. – 2011-12-30
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0Then what would be correct? – 2011-12-30
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1$(AB)^T = B^T A^T$ -- matrices multiply by juxtaposition. – 2011-12-30
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2What about this? http://www.proofwiki.org/wiki/Transpose_of_Matrix_Product – 2011-12-30
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0@ Martin: Wonderful! Thanks for the website, I often need simple proofs like this one (but just complicated enough for me, I'm really not much of a mathematician), I imagine lots of them are on there. – 2011-12-30
4 Answers
The $(i,j)$ entry of $(AB)^T$ is the $(j,i)$ entry of $AB$, $\sum \limits_k a_{jk}b_{ki}$. The $(i,j)$ entry of $B^TA^T$ is $\sum \limits_k b_{ki}a_{jk}$ (the $i$th row of $B^T$ is the $i$th column of $B$, the $j$th column of $A^T$ is the $j$th row of $A$).
Just a tip, you should remember and understand that the $(i,j)$ entry of multiplying $AB$ is $\sum \limits_k a_{ik}b_{kj}$. As $k$ advances, you are going across row $i$ of $A$ and down column $j$ of $B$.
Unfold the definitions of matrix multiplication and transposition in both cases. Notice that each element of the matrix on the two sides of the equals signs end up being the same sum of products of elements of $A$ and $B$.
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1Could you give a worked out example, I mess up with the indexes. – 2011-12-30
For what's it's worth:
$$ \overbrace{\left[\matrix{a_{11}&a_{12}&a_{13} \cr \color{maroon}{a_{21}}&\color{maroon}{a_{22}}&\color{maroon}{a_{23}} \cr a_{31}&a_{32}&a_{33} \cr }\right]}^A \overbrace{ \left[\matrix{b_{11}&b_{12}&\color{maroon}{b_{13}} \cr b_{21}&b_{22}&\color{maroon}{b_{23}} \cr b_{31}&b_{32}&\color{maroon}{b_{33}} \cr }\right]}^B = \underbrace{ \left[\matrix{ \cdot & \cdot & \cdot \cr \cdot & \cdot & \color{maroon}{ a_{21}b_{13}+ a_{22}b_{23}+ a_{23}b_{33} } \cr \cdot & \cdot & \cdot \cr }\right]}_{AB} $$ $$ \phantom{ \overbrace{\left[\matrix{a_{11}&a_{12}&a_{13} \cr \color{maroon}{a_{21}}&\color{maroon}{a_{22}}&\color{maroon}{a_{23}} \cr a_{31}&a_{32}&a_{33} \cr }\right]}^A \overbrace{ \left[\matrix{b_{11}&b_{12}&\color{maroon}{b_{13}} \cr b_{21}&b_{22}&\color{maroon}{b_{23}} \cr b_{31}&b_{32}&\color{maroon}{b_{33}} \cr }\right]}^B=} \underbrace{ \left[\matrix{ \cdot & \cdot & \cdot \cr \cdot & \cdot & \cdot \cr \cdot & \color{maroon}{ a_{21}b_{13}+ a_{22}b_{23}+ a_{23}b_{33} } & \cdot \cr }\right]}_{[AB]^T} $$ $$ \underbrace{ \left[\matrix{b_{11}&b_{21}&b_{31} \cr {b_{12}}&{b_{22}}&{b_{32}} \cr \color{maroon}{b_{13}}&\color{maroon}{b_{23}}&\color{maroon}{b_{33}} \cr }\right]}_{B^T} \underbrace{ \left[\matrix{a_{11}&\color{maroon}{a_{21}}& {a_{31}} \cr a_{12}&\color{maroon}{a_{22}}& {a_{32}} \cr a_{13}&\color{maroon}{a_{23}}& {a_{33}} \cr }\right]}_{A^T} = \underbrace{ \left[\matrix{ \cdot & \cdot & \cdot \cr \cdot & \cdot & \cdot \cr \cdot & \color{maroon}{ a_{21}b_{13}+ a_{22}b_{23}+ a_{23}b_{33} } & \cdot \cr }\right]}_{B^TA^T} $$
Let $r_1,.., r_n$ be the rows of $A$ and $c_1,..,c_n$ be the columns of $B$.
The $ij$ entry in $(AB)^T$ is the $ji$ entry in $AB$, which is the dot product $r_j \cdot c_i$.
The $ij$ entry in $B^TA^T$ is the dot product between the $i$th row of $B^T$ and $j$th column of $A^T$, thus the dot product between the $i$th column of $B$ and $j$th row of $A$, hence $c_i \cdot r_j$.
Since
$$r_j\cdot c_i =c_i \cdot r_j $$
we get
$$(AB)^T=B^TA^T \,.$$