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Einstein's summation convention

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Old   February 23, 2007, 15:28
Default Einstein's summation convention
  #1
Lionel S.
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Hi all,

I found a strange consistency leak in summation convections: Suppose I have two matrix, A and B

tr(AB) = (AB)_ii

Since

(AB)_ij = A_im B_mj

Then

tr(AB) = A_im B_mi

But if A is symmetric it means that

tr(AB) = A_mi B_mi

Which is equal to A:B, the scalar (inner) tensorial product. Which is impossible because

tr(AB) not equal to A:B even A or B are symmetric

There is a fault somewhere, but I can't find it.

Do you have an idea ? Thanx in advance...

Bye Lionel

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Old   February 23, 2007, 16:09
Default Re: Einstein's summation convention
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Praveen. C
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Its correct. Just work out a case of 2x2 matrices to convince yourself.
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Old   February 23, 2007, 16:28
Default Re: Einstein's summation convention
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Lionel S.
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Yes ... surprisingly, it works, I didn't believe it first.

... Thanx, it could help for one of my next exams (mechanics of composite materials) ;-)
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