i) 1 Let A, B E Man (C) be two square matrices. 14 A~ B. Explain what it means that the matrices A and B are similar, i.e., Prove that if the matrices A and B are similar, then the the the set of of the matrix A is the same as the set of eigenvalues of the matrix B. Give an example of two matrices A, B € M2,2(C) such that eigenvalues A) A is not similar to B; and B) A and B have the same the same set of eigenvalues. Give a detailed justification to your answer.
i) 1 Let A, B E Man (C) be two square matrices. 14 A~ B. Explain what it means that the matrices A and B are similar, i.e., Prove that if the matrices A and B are similar, then the the the set of of the matrix A is the same as the set of eigenvalues of the matrix B. Give an example of two matrices A, B € M2,2(C) such that eigenvalues A) A is not similar to B; and B) A and B have the same the same set of eigenvalues. Give a detailed justification to your answer.
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
Needed to be solved Part B only
Correctly in 15 minutes and get the thumbs up please show neat and clean work for it
![3. a) [
i) [
iii) 1-
iv)
b) .
c) [ª
such that
Let x = (1, 2) € R² and y = (y₁, y2) € R².
Find a symmetric matrix A € M2,2 (R) such that
x Ax= -x + 4x1x₂x².
- Find a diagonal matrix D E M2,2 (R) and a rotation matrix Q € M2,2 (R)
ii) -
1-
A~ B.
Explain why the equation
is an equation of a hyperbola.
s] Let up and u₂ be the tangent vectors to the asymptotes of the hyperbola
above (see fig. 1).
x Ax=y Dy, where y = Q¹x.
A) Find the canonical (y₁, 92) coordinates of the vectors u₁ and u₂.
B)
For the same vectors u₁ and u2, find the original (21, 2) coordinates of the
vectors up and u₂.
-1 Let A, B = Mn (C) be two square matrices.
eigenvalues
x Ax = 1
Explain what it means that the matrices A and B are similar, i.e.,
Prove that if the matrices A and B are similar, then the the the set of
of the matrix A is the same as the set of eigenvalues of the matrix B.
Give an example of two matrices A, B E M2,2(C) such that
PROPE
UNS
A) A is not similar to B; and
B) A and B have the same the same set of eigenvalues.
Give a detailed justification to your answer.
Let a R and let
Qa =
sin a cos a
sin a
cos a
€ M₂,2 (R)
Prove that Qa is orthogonal.
Let x € R². Prove that ||Qax| = |x||. Make sure that you carefully
state on the margins of your work, every trigonometric identity and/or any other
result you use in your proof.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2e672a1-54a8-4743-aad6-f3d1c4a0c4ab%2Ffd9bedf0-024f-4789-9ae0-70fd44694cca%2Fxc32xv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. a) [
i) [
iii) 1-
iv)
b) .
c) [ª
such that
Let x = (1, 2) € R² and y = (y₁, y2) € R².
Find a symmetric matrix A € M2,2 (R) such that
x Ax= -x + 4x1x₂x².
- Find a diagonal matrix D E M2,2 (R) and a rotation matrix Q € M2,2 (R)
ii) -
1-
A~ B.
Explain why the equation
is an equation of a hyperbola.
s] Let up and u₂ be the tangent vectors to the asymptotes of the hyperbola
above (see fig. 1).
x Ax=y Dy, where y = Q¹x.
A) Find the canonical (y₁, 92) coordinates of the vectors u₁ and u₂.
B)
For the same vectors u₁ and u2, find the original (21, 2) coordinates of the
vectors up and u₂.
-1 Let A, B = Mn (C) be two square matrices.
eigenvalues
x Ax = 1
Explain what it means that the matrices A and B are similar, i.e.,
Prove that if the matrices A and B are similar, then the the the set of
of the matrix A is the same as the set of eigenvalues of the matrix B.
Give an example of two matrices A, B E M2,2(C) such that
PROPE
UNS
A) A is not similar to B; and
B) A and B have the same the same set of eigenvalues.
Give a detailed justification to your answer.
Let a R and let
Qa =
sin a cos a
sin a
cos a
€ M₂,2 (R)
Prove that Qa is orthogonal.
Let x € R². Prove that ||Qax| = |x||. Make sure that you carefully
state on the margins of your work, every trigonometric identity and/or any other
result you use in your proof.
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