[a b (e) Let B = be any 2 x 2 matrix. [cos a = u11 [sin a (i) Show that there are real numbers un and a such that Hint: erpress as a scalar multiple of a unit vector, and hence find an erpression for u in terms of a and c. (ii) Let a e R. Use the invertibility of R, to prove that there are unique U12, U22 € R such that -- sin a cos a [cos a u12 + u22 (sino [sin a] (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R,U for some a €R and some upper-triangular matrix U. (iv) Suppose that B = RU = R3V, where a, B e R and U and V are upper- triangular. Prove that if B is invertible, then U = ±V. %3D
[a b (e) Let B = be any 2 x 2 matrix. [cos a = u11 [sin a (i) Show that there are real numbers un and a such that Hint: erpress as a scalar multiple of a unit vector, and hence find an erpression for u in terms of a and c. (ii) Let a e R. Use the invertibility of R, to prove that there are unique U12, U22 € R such that -- sin a cos a [cos a u12 + u22 (sino [sin a] (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R,U for some a €R and some upper-triangular matrix U. (iv) Suppose that B = RU = R3V, where a, B e R and U and V are upper- triangular. Prove that if B is invertible, then U = ±V. %3D
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 27EQ
Related questions
Question
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![(e) Let B =
be any 2 x 2 matrix.
COS a
(i) Show that there are real numbers un and a such that
u11
sin a
Hint: erpress
as a scalar multiple of a unit vector, and hence find an
erpression for un in terms of a and c.
(ii) Let a € R. Use the invertibility of R, to prove that there are unique
u12, U22 €R such that
cos a
= u12
- sin a
+ u22
cos a
sin a
(iii) Use parts (i) and (ii) to show that B can be expressed in the form
B = R,U
for some a €R and some upper-triangular matrix U.
(iv) Suppose that B = R,U = R3V, where a, B eR and U and V are upper-
triangular. Prove that if B is invertible, then U = ±V.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb4af5bf-9b75-4cca-a55b-ef482e95bc73%2F1e7a9309-be3d-4d13-be4f-ecb3644f045e%2F9ftv4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(e) Let B =
be any 2 x 2 matrix.
COS a
(i) Show that there are real numbers un and a such that
u11
sin a
Hint: erpress
as a scalar multiple of a unit vector, and hence find an
erpression for un in terms of a and c.
(ii) Let a € R. Use the invertibility of R, to prove that there are unique
u12, U22 €R such that
cos a
= u12
- sin a
+ u22
cos a
sin a
(iii) Use parts (i) and (ii) to show that B can be expressed in the form
B = R,U
for some a €R and some upper-triangular matrix U.
(iv) Suppose that B = R,U = R3V, where a, B eR and U and V are upper-
triangular. Prove that if B is invertible, then U = ±V.
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