linear algebra question

**kingsolomonsgrave** · February 26th 2013, 08:10 AM

in the question why can you change the order of AB and BA in

5(AB^T)^T = 5(B^T)^T A^T ?

where ^T means the transpose of a matrix

**hairymclairy** · February 26th 2013, 03:55 PM

This is a standard property of transposition ( $(AB)^T=B^TA^T$ ). Let $A$ be an $n\times m$ matrix and let $B$ be an $m\times n$ matrix. Now consider their transposes $A^T$ and $B^T$ . Row $i$ of $A$ is exactly the same as column $i$ of $A^T$ , similarly column $i$ of $B$ is exactly the same as row $i$ of $B^T$ . So you should be able to see that

$row_i(A)\cdot column_j(B)=row_j(B^T)\cdot column_i(A^T),$

so it follows that $(AB)^T=B^TA^T$ .

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