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Quadratic Forms & Lagrange Multipliers

  1. Mar 6, 2012 #1

    ElijahRockers

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    Gold Member

    1. The problem statement, all variables and given/known data

    I'm having trouble grasping this. Our teacher has decided to combine elements from Linear Algebra, and understanding Quadratic forms with our section on lagrange multipliers. I am barely able to follow his lectures. If I look down to take notes, even once, by the time I look back up I no longer understand what he's talking about. It is frustrating, but I am determined.

    Anyway, I did the first part, which is to construct three quadratic equations. That part is easy. I also 'understand' (that is a strong word, maybe 'remember' is better) how to find the eigenvectors. Part e) is what gives me the most trouble. I have never seen that formula, and I have no idea how it works. The left hand side is a column, and the right hand side is like a normal expression. What does that mean?

    I went to his office and he blazed through it, showing me some alleged "shortcut" involving matrices that seemed to take a suspiciously long time, and just confused me more. I still don't get it.

    Any help at all would be appreciated.


    2. Relevant equations



    3. The attempt at a solution

    For part 2, grad_Q=t*grad_g
    so

    (2-t)x+y=0
    x+(2-t)y=0
    x2+y2=1

    The characteristic polynomial is
    t2-4t+3=0

    so t1=3 and t2=1

    Therefore Q can be classified as positive (since both eigenvalues are positive)

    For t=3,
    y=x=(√2'/2)

    For t=1,
    y=-x=(-√2'/2)

    So eigenvector v1 = <√2'/2 , √2',2>
    v2 = <√2'/2 , -√2'/2>

    I'm lost at this point. I have two vectors now, that cross through the origin, are perpendicular, and touch the unit circle in four places, but I don't really see how these vectors relate to the quadratic equation, and what they mean, geometrically. Same with the eigenvalues.

    None of this is in my textbook besides the lagrange multipliers, and my classmates are just as lost as I am.

    please help! Thanks in advance! :)
     
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