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Complex Scalar Product Proof

  • #1

Homework Statement



This is what we are given in the assignment:

Recall a definition of scalar product on complex numbers. Let A = [[3,1],[1,2]]. Prove that the product as defined by:

* => dot product

u * v := uT * A * conjugate(v)

( = Sum from i,j=1 to 2; uiAijconjugate(vj) )

is a scalar product on C according to the definition.

Homework Equations



We are give that the following equation will be useful:

2(ac) >= -a2 -c2 for all a,c as elements of R

The Attempt at a Solution



There are many of us working on this and we were not even sure exactly how to start this problem. It is trivial to prove the Sum given is equal to uT * A * conjugate(v). But from there we weren't sure exactly what else to prove.

Any help would be extremely appreciated
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
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Have you written out the computations?
For
[tex]u= \begin{bmatrix}u_1 \\ u_2\end{bmatrix}[/tex]
[tex]v= \begin{bmatrix}v_1 \\ v_2\end{bmatrix}[tex]

The product is
[tex]u*v= \begin{bmatrix}u_1 & u_2\end{bmatrix}\begin{bmatrix}3 & 1 \\ 1 & 2\end{bmatrix}\begin{bmatrix}\overline{v_1} \\ \overline{v_2}\end{bmatrix}= (3u_1+ u_2)\overline{v_1}+ (u_1+ 3u_2)\overline{v_2}[/tex]

Now what are the requirements for a scalar product- what is the definition? Does this satisfy those requirements?
 
  • #3
Have you written out the computations?
For
[tex]u= \begin{bmatrix}u_1 \\ u_2\end{bmatrix}[/tex]
[tex]v= \begin{bmatrix}v_1 \\ v_2\end{bmatrix}[tex]

The product is
[tex]u*v= \begin{bmatrix}u_1 & u_2\end{bmatrix}\begin{bmatrix}3 & 1 \\ 1 & 2\end{bmatrix}\begin{bmatrix}\overline{v_1} \\ \overline{v_2}\end{bmatrix}= (3u_1+ u_2)\overline{v_1}+ (u_1+ 3u_2)\overline{v_2}[/tex]

Now what are the requirements for a scalar product- what is the definition? Does this satisfy those requirements?
Yes I got this far, my only problem is that I don't see any logical next step, I may be missing something very obvious, but even after reading more about the definition of scalar product I don't understand what more there is to prove

And I also do not understand the purpose of the matrix A
 

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