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August 23rd, 2015, 06:09 AM   #1
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Understanding "trivial" proof

Hi,

I am reading this paper and on page 7 at the bottom there is a "Theorem 1" with a trivial proof that I don't understand.

In particular, why are these equal?
<J J^T e, e> = <J^T e, J^T e>

(<> is the dot product.)
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August 23rd, 2015, 06:25 AM   #2
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Do you know the definitions of any of these things?
In particular, what is the definition of "dot product", <A, B>?
What is the definition of "J^T" ($\displaystyle J^T$)?

(Does your textbook actually say "dot product" and not "inner product'?)
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August 23rd, 2015, 07:41 AM   #3
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Quote:
Originally Posted by Country Boy View Post
Do you know the definitions of any of these things?
In particular, what is the definition of "dot product", <A, B>?
$\displaystyle A \cdot B = \sum_{i} (A_i * B_i)$
Quote:
Originally Posted by Country Boy View Post
What is the definition of "J^T" ($\displaystyle J^T$)?
$\displaystyle J^T$ is the transpose of $\displaystyle J$. It is a matrix where the rows are the columns of $\displaystyle J$.

$\displaystyle J^T_{ij} = J_{ji}$
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Originally Posted by Country Boy View Post
(Does your textbook actually say "dot product" and not "inner product'?)
No. I was probably wrong and <> meant the inner product. The definition of the inner product seems to be more abstract with axioms like "conjugate symmetry", "linearity in the first argument" and "positive-definitness". Still not sure how to apply this.
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August 23rd, 2015, 03:08 PM   #4
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In a text where they talk about an "inner product", meaning a binary operation defined on a vector space, such that:
1) <au+ bv, w>= a<u, w>+ b<v, w>
2) <u, v>= <v, u> if the vector space is over the real numbers, <u, v>= <v, u>* (the complex conjugate) if over the complex numbers
3) <v, v>>= 0 and <v, v>= 0 if and only if v= 0

would be more likely to define A* by <Au, v>= <u, A*v> for all u and v than in terms of "matrices".

I suspect that you have learned some basic "vector" and "matrix" properties and are trying to interpret something much more abstract in terms of those. For one thing, "matrices" and "dot products" can only be defined over finite dimensional vector spaces.
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