How Does the Change of Coordinate Matrix Transform an Ellipse Equation?

[jeff1evesque]

In geometry the change of variable,

x = (2 / sqrt(5))x' - (1 / sqrt(5))y' (#1)
y = (1 / sqrt(5))x' + (2 / sqrt(5))y' (#2)
can be used to transform the equation 2x^2 - 4xy + 5y^2 = 1 into the simpler equation (x')^2 + 6(y')^2 = 1, in which form it is easily seen to be the equation of an ellipse.

B and B' are the standard ordered basis and new rotated basis respectively

My question:
Why is ^B_B' have such a representation with B and B'? Why wouldn't it be ^B'_B?

 

[Cantab Morgan]

Why is ^B_B' have such a representation with B and B'? Why wouldn't it be ^B'_B?

Can you try to state your question differently? I was following what you wrote up to that point, but I'm not familiar with the notation you're using at the end.

 

[jeff1evesque]

Can you try to state your question differently? I was following what you wrote up to that point, but I'm not familiar with the notation you're using at the end.

So in this particular example,
B' = { 1/sqrt(5)(2, 1), 1/sqrt(5)(-1, 2) }
Question: what about the basis B, what is it's values?

Geometrically the change of variable is (x,y) --> (x',y') is a change in the way that the position of a point P in the plane.

The change of variable is actually a change from [P]_B = (x, y), the coordinate vector of the point P relative to the ordered basis B = {e1, e2}, to [P]_B' = (x', y'), the coordinate vector of P relative to the new rotated basis B'.

Notice also that the matrix

| 2 -1 |​

Q = 1/sqrt(5) | 1 2 |

equals <i>^B_B&#039;</i>, where I denotes the identity transformation on R^2. Thus [v]_B = Q[v]_B_&#039; for all v in R^2. A similar result is true in general.

Two questions for the last two sentences from above:
1.) What exactly is [v]_B? What are the values for this vector, how is it obtained?
2.) What do they mean by a similar result is true?

thanks,


JL

 

[Cantab Morgan]

<i>^B_B&#039;</i>

Sorry, I still don't know what that notation means.

But here's what I see is happening. The ellipse 2x^2 -4xy + 5y^2 = 1 represents a quadratic form, meaning a particular inner product on R^2. Imagine that there is a symmetric matrix S with positive eigenvalues...

<br /> S = \left( \begin{array}{cc} 2 &amp; -2 \\ -2 &amp; 5 \end{array} \right)<br />

Then we're looking at the set of all points v = \left( \begin{array}{c} x \\ y \end{array} \right) satisfying Sv \cdot v = 1. These form an ellipse. By finding the eigenvectors of that matrix S, and normalizing them, we assemble your rotation matrix Q. Q rotates the plane. Consider its transpose and see that v&#039; = Q^Tv, and we get the ellipse x&#039;^2 +6y&#039;^2 = 1. Note that these coefficients 1 and 6 are the eigenvalues of S, and they represent the maximum and minimum values of the ellipse's distance from the origin.

So, what is v? This question I understand. What you are calling v_B. Well, it can really be any point in the plane that you transform by Q^T, but in our case we can limit it to those points in the plane on the ellipse. That is, all points for which Sv \cdot v = 1.

I don't understand your question about the values of the basis B. You wrote down the basis B.

Like you, I also don't know what they mean by "a similar result is true".