Inner Product Question

  • Thread starter Onezimo Cardoso
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  • #1
Onezimo Cardoso

Homework Statement


upload_2017-7-28_18-15-22.png


Homework Equations


upload_2017-7-28_18-15-30.png


The Attempt at a Solution


[/B]
I could show that the first of the three properties are valid for any value of a,b,c but I couldn’t find a way to show the forth one.

Follow all the procedures I already did:

upload_2017-7-28_18-18-29.png


upload_2017-7-28_18-18-46.png

upload_2017-7-28_18-19-33.png
 

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Answers and Replies

  • #2
Ray Vickson
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Homework Statement


View attachment 207920

Homework Equations


View attachment 207921

The Attempt at a Solution


[/B]
I could show that the first of the three properties are valid for any value of a,b,c but I couldn’t find a way to show the forth one.

Hint: if ##{\bf x} = (x_1,0)##, it is easy to show that ##\langle {\bf x,x} \rangle > 0## for any ##x_1 \neq 0##. Also: for ##x_2 \neq 0## we have ##(x_1,x_2) = x_2(x_1/x_2,1)##, so ##\langle {\bf x,x} \rangle = x_2^2 \langle {\bf u,u} \rangle##, where ##{\bf u} = (x_1/x_2,1) \equiv (t,1)##.
 
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  • #4
StoneTemplePython
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are you allowed to use spectral theory here?

Your matrix

##
\begin{bmatrix}
a & b\\
b & c
\end{bmatrix}##

is real symmetric. If the determinant is positive (one of your conditions in the iff) and the trace is positive (implied by ##a \gt 0##... why?) then it this tells you...
 
  • #5
Onezimo Cardoso
upload_2017-7-31_21-22-7.png


upload_2017-7-31_21-22-39.png


upload_2017-7-31_21-23-18.png
 

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  • #6
ehild
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The Attempt at a Solution


[/B]
I could show that the first of the three properties are valid for any value of a,b,c but I couldn’t find a way to show the forth one.

Follow all the procedures I already did:
If you factorize Δ you get Δ=4x22(b2-4ac)
 
  • #7
Ray Vickson
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