Eigenvalues and eigenvectors are useful in sketching complicated graph of conic sections. We will focus on the equation in below form: ax² +2bxy+ay2² = = c with b 0 and c > 0. b (8). b a Let (x, y) = ax² + 2bry + ay2 and denote A = Note that: V(x,y) = (x v) 4 (4) = ( (4), 4 (*)) 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 u, 7 of R2, formed by eigenvectors of A. Then, express a fixed but unknown vector () ER2 as a linear combination of u and v. (c) Use result from (b) to transform V(x, y) = ((*), 4 (1)) some u(x, y) and v(x, y), with a, ß ER being eigenvalues of A. into (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+ay2² = = c with b 0 and c > 0. b (8). b a Let (x, y) = ax² + 2bry + ay2 and denote A = Note that: V(x,y) = (x v) 4 (4) = ( (4), 4 (*)) 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 u, 7 of R2, formed by eigenvectors of A. Then, express a fixed but unknown vector () ER2 as a linear combination of u and v. (c) Use result from (b) to transform V(x, y) = ((*), 4 (1)) some u(x, y) and v(x, y), with a, ß ER being eigenvalues of A. into (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
Related questions
Question
Transcribed Image Text:(d) Use result from (c) to sketch 2x² + 2xy + 2y² = 3 based on the coordinate system formed
by u and as the axes.
Hint: You might want to revise on sketching ellipse.
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:
ar² +2bry + ay²: = c with b 0 and c > 0.
(2).
b
Let V(x, y)
=
ax² +2bxy + ay2 and denote A =
Note that:
(x, y) = (x y) A
¹() = (()₁ ^ ())
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 u, v of R2, formed by eigenvectors of A. Then, express a
fixed but unknown vector
(₁)
ER2 as a linear combination of u and v.
(c) Use result from (b) to transform V(x, y) = ( (*), 4 (”))
A
some u(x, y) and v(x, y), with a, B E R being eigenvalues of A.
into (u, v) au² + Bv² for
=
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