Eigenvalues and eigenvectors are useful in sketching complicated graph of conic sections. We will focus on the equation in below form: ax² + 2bxy+ay² = c with b 0 and c> 0. a (öö). b Let V(x, y) = ax² +2bry + ay² and denote A = Note that: v (2, 1) = (2 v) A (*) = ((*), 4 (*)) А where (,): R2 x R2 → R is the standard inner (dot) product on R². (a) Determine all eigenvalues of A, and their corresponding eigenspaces. (b) Construct an orthonormal basis ü, 7 of R2, formed by eigenvectors of A. Then, express a fixed but unknown vector (+) ER2 as a linear combination of u and . (c) Use result from (b) to transform (z,y) = ((*). 4 ()) A some u(x, y) and v(x, y), with a, ß ER being eigenvalues of A. into V(u, v) = au² + Bv² for
Eigenvalues and eigenvectors are useful in sketching complicated graph of conic sections. We will focus on the equation in below form: ax² + 2bxy+ay² = c with b 0 and c> 0. a (öö). b Let V(x, y) = ax² +2bry + ay² and denote A = Note that: v (2, 1) = (2 v) A (*) = ((*), 4 (*)) А where (,): R2 x R2 → R is the standard inner (dot) product on R². (a) Determine all eigenvalues of A, and their corresponding eigenspaces. (b) Construct an orthonormal basis ü, 7 of R2, formed by eigenvectors of A. Then, express a fixed but unknown vector (+) ER2 as a linear combination of u and . (c) Use result from (b) to transform (z,y) = ((*). 4 ()) A some u(x, y) and v(x, y), with a, ß ER being eigenvalues of A. into V(u, v) = au² + Bv² for
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:3. Eigenvalues and eigenvectors are useful in sketching complicated graph of conic sections. We
will focus on the equation in below form:
ax² +2bxy+ay² = c with b #0 and c> 0.
a
(öd).
b
Let V(x, y) = ax² + 2bry + ay2 and denote A =
Note that:
V(x, y) = (x y) A ^ (*) = ((*), ^ (*))
where (,): R2 x R2 → R is the standard inner (dot) product on R².
(a) Determine all eigenvalues of A, and their corresponding eigenspaces.
(b) Construct an orthonormal basis ü, ü of R2, formed by eigenvectors of A. Then, express a
fixed but unknown vector ER2 as a linear combination of u and .
(;)
(e) Use result from (b) to transform V(x, y) = ((*). 4 ()) into V(u, v) au²+ Bu² for
some u(x, y) and v(x, y), with a, ß ER being eigenvalues of A.
Transcribed Image Text:(d) Use result from (c) to sketch 22² + 2xy + 2y²:
= 3 based on the coordinate system formed
by and as the axes.
Hint: You might want to revise on sketching ellipse.
16
a
mustory board game players move between locations
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