Question

Using the Spectral Theorem.

Problem 4. Sketch a plot of the ellipse 37a2 - 18ry + 13y2 -100. What are the directions of the principal axes? (Note: the problem set originally used a different ellipse, but this one will be cleaner. You can use either.)

Answer 1

skech of given ellipse is

" ellipse is 37x^2 - 18xy + 13y^2 = 100 Its plot is shown in figure Now spectral theorem states that "" if a Matrix M such that M = M^T (transpose of M) then M is diagonalizable matrix i.e, exist matrix P such that p^-1 MP is diagonal matrix "" here let Q (x, y) = 37x^2 - 18xy + 13y^2 a quadratic form which can be written as Q (x, y) = [x y] [37 -9 -9 13] [x y] Let vartheta vec = [x y] = (x y)^T Then becomes Q (x, y) = vartheta vec^T [37 -9 -9 13] vartheta vec Let M = [37 -9 -9 13] Rightarrow Q (x, y) = vartheta vec M vartheta vec clearly M is symmetric 2 times"