- #1
CheMech
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Homework Statement
http://imageshack.us/photo/my-images/861/screenshot20111211at928.png/
I am only concerned with part (a) of this problem.
Homework Equations
This question was assigned in a linear algebra class while we were learning about eigenvalues and eigenvectors (I have since finished the class, but since this problem was assigned in the last week of the class before finals, I wasn't able to find enough time to go get help for this question).
The Attempt at a Solution
I learned in a previous calculus course that the gradient vector points in the direction of fastest increase, and that the negative of the gradient vector points in the direction of the greatest decrease. I figured I would use this as a check after solving it using the linear algebra method (as was the way this problem was "supposed" to be solved in this class).
So, I tried finding the gradient:
f(x, y) = 3x2 + 6xy - 5y
and...
[itex]\nabla[/itex]f = < 6x + 6y, 6x - 10y2 >
However, evaluating this at (0, 0, 0) to determine the direction of greatest change yields <0,0>.
The actual answer (the linear algebra way) is that the directions of fastest increase/decrease correspond to the directions of the eigenvectors of the matrix that represents the given quadratic form.* I have a few issues with this:
1) Why do the eigenvectors' directions correspond to the direction of fastest increase/decrease? I learned before that the direction of the gradient pointed in the direction of the fastest increase (and that the negative of it pointed in the direction of the fastest decrease).
2) Looking at part (b) of this question: Based on what I said above, I would say that the angle between the two directions is 180 degrees, because the gradient and its opposite are opposite of one another. However, this turns out not to be the case according to the actual answer stated in 1).Any help at all is appreciated. This question has been bugging me :P* This:
\begin{bmatrix}
3 & 3 \\
3 & -5
\end{bmatrix} is the matrix of the quadratic form given.
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