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Transpose of Matrix Product - Property

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joniengr

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Hi,
I guess the transpose of product property is only for square matrix when rows and columns are equal in A and B. For example 2 x 2 and 3 x 3 Matrix. I think this is not true otherwise when A is 2 x 3 and B is 3 x 2. Can someone confirm this please ?

(AB)⊺=B⊺A⊺
 

albbg

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No it's always true.
If we have two matrix A with indexes i=1..m, k=1..p and B with indexes k=1..p, j=1..n
the transpose of the product is given by:

\[{[AB]}^{T}\]=\[\sum_{k = 1}^p\]A(i,k)B(k,j)

we can commutate A(i,k) and B(k,j) that is:

\[{[AB]}^{T}\]=\[\sum_{k = 1}^p\]B(k,j)A(i,k)

remembering now that \[{A(i,k)}^{T}\]=A(k,i), by definition of transpose we can write:

\[{[AB]}^{T}\]=\[\sum_{k = 1}^p\]\[{B(j,k)}^{T}\]\[{A(k,i)}^{T}\]
 

joniengr

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Hi,

If there are two Matrix with dimension A (2 x 3 ) and B (3 x 2). The product will be of size (2 x 2). The transpose will also be of dimension (2x2). This will be the left hand side of (AB)⊺=B⊺A⊺

Solving for right hand side, if I take transpose of A and B then the dimension of resultant matrix will be AT (3 x 2) and BT (2 x 3). The product of them will be of dimension (3 x 3). How this will be equal to the left hand side ?
 

albbg

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What you say is not correct: AT(3x2) and BT(2x3), then BT(2x3)*AT(3x2) = Matrix (2x2)
 

    joniengr

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