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March 29, 2024, 06:39:29 pm

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m@tty

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Help please :)
« on: March 15, 2010, 05:14:16 pm »
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I need help with two questions(of many which I don't understand). All assistance is appreciated :)
These are questions on matrices.



Expand to show that 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 , where and are both matrices. I found the trace to be . 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
It follows that


And these two differ only in the name of the variable, which is arbitrary. Hence,

What should be written for this last part, to show that ?
Or can I simply state that they are equal?

Thanks :)
« Last Edit: March 15, 2010, 09:05:21 pm by m@tty »
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m@tty

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Re: Help please :)
« Reply #1 on: March 15, 2010, 09:09:49 pm »
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For the first one I can't think of any linking factors, beside the common matrix, . But that doesn't even help. It's probably a simple property or something I overlooked, sigh...
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Mao

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Re: Help please :)
« Reply #2 on: March 16, 2010, 01:16:39 am »
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Expand to show that is always symmetric. Why does the same proof show that is symmetric?

A matrix B is symmetric if

(if its transpose is the same as itself, it must be symmetric)

In this case, let , by showing , you have effectively shown that , hence must be symmetric.
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Mao

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Re: Help please :)
« Reply #3 on: March 16, 2010, 01:27:19 am »
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What should be written for this last part, to show that ?

The equality is inherently true, since you have just swapped the position of the two variables i and k.
« Last Edit: March 16, 2010, 01:33:15 am by Mao »
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m@tty

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Re: Help please :)
« Reply #4 on: March 16, 2010, 04:37:19 pm »
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Thanks Mao. :)

Expand to show that is always symmetric. Why does the same proof show that is symmetric?

A matrix B is symmetric if

(if its transpose is the same as itself, it must be symmetric)

In this case, let , by showing , you have effectively shown that , hence must be symmetric.

How does that explain why the same proof shows that is symmetric?
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Re: Help please :)
« Reply #5 on: March 16, 2010, 05:10:41 pm »
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Let to show that is symmetric.
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m@tty

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Re: Help please :)
« Reply #6 on: March 16, 2010, 05:19:57 pm »
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So which is symmetric?

How does showing this 'explain why the same proof shows that is symmetric'?
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Re: Help please :)
« Reply #7 on: March 16, 2010, 07:24:04 pm »
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Let and repeat the proof with B:



Now plug A back in.

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Re: Help please :)
« Reply #8 on: March 16, 2010, 11:46:13 pm »
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Let and repeat the proof with B:



Now plug A back in.

So, the original proof proved that matrices of the form are always symmetric. Therefore is symmetric, but , and the symmetry found before still holds. Is this it?
I think it is, thankyou, everyone :) :) :)

:)
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