I need help with two questions(of many which I don't understand). All assistance is appreciated
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These are questions on matrices.
Expand
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?(A^TA)^T)
to show that
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?A^TA)
is always symmetric.
Why does the same proof show that
is symmetric?I don't know what to say here, I could perform the same test and prove that it is symmetric, but that isn't using 'the same proof' is it?
In the second question we are told to deduce the trace of an arbitrary matrix
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?AB)
, where
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?A)
and
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?B)
are both
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?n\times n)
matrices. I found the trace to be
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\sum_{i=1}^n\sum_{k=1}^n a_{ik}b_{ki})
. The second part of the question requires us to
show that the traces of
and
are equal.My steps were as follows:
From above:
Trace
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?AB \ =\sum_{i=1}^n\sum_{k=1}^n a_{ik}b_{ki})
It follows that
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\begin{align} trace(BA)&=\sum_{i=1}^n\sum_{k=1}^n b_{ik}a_{ki} \\&=\sum_{i=1}^n\sum_{k=1}^n a_{ki}b_{ik} \\ &=\sum_{k=1}^n\sum_{i=1}^n a_{ki}b_{ik} \end{align})
And these two differ only in the name of the variable, which is arbitrary. Hence,
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?trace(BA)=trace(AB))
What should be written for this last part, to show that
?Or can I simply state that they are equal?
Thanks
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