Quadratic Forms & Lagrange Multipliers

[ElijahRockers]

Homework Statement

I'm having trouble grasping http://www.math.tamu.edu/~vargo/courses/251/HW6.pdf. 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.

Homework Equations

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! :)

 

[mighty2000]

Dear student,

I understand your frustration and confusion with this topic. It can be difficult to grasp new concepts, especially when they are combined with other topics. First of all, let me explain the formula you are struggling with, grad_Q=t*grad_g. This formula is known as the gradient descent method and it is commonly used in optimization problems. In this case, we are using it to find the minimum or maximum values of a quadratic function subject to a constraint (given by the function g). The left-hand side of the equation represents the gradient of the quadratic function Q, while the right-hand side represents the gradient of the constraint function g multiplied by a scalar t. This method helps us find the optimal values of t that satisfy both equations simultaneously.

Moving on to part e), you have correctly found the eigenvalues and eigenvectors of the quadratic function Q. The eigenvalues tell us about the nature of the function, whether it is positive or negative, and the eigenvectors represent the directions in which the function does not change. In this case, since both eigenvalues are positive, Q can be classified as positive definite. The eigenvectors you have found represent the directions along which Q does not change, and they are perpendicular to each other (since they are the eigenvectors of a symmetric matrix).

To better understand the geometric interpretation of these vectors, you can plot them on a graph. The eigenvectors will form the axes of an ellipse, and the eigenvalues will determine the size of the ellipse. This will give you a better understanding of how the quadratic function behaves in different directions.

I hope this helps in your understanding of this topic. If you still have any doubts or questions, do not hesitate to ask your teacher or seek help from a peer or tutor. Understanding these concepts will not only help you in this assignment but also in your future studies and research.

Best of luck!