(e) Let B be 2 x 2 matrix. any a CO a = Uj1 (i) Show that there are real numbers u11 and a such that sin a a Hint: express as a scalar multiple of a unit vector, and hence find an expression for u11 in terms of a and c. ( ii ) Let α R. Use the invertibility of Ra to prove that there are unique U12, U22 E R such that co a sin a U12 sin a + U22 COs a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = RQU for some a E R and some upper-triangular matrix U. (iv) Suppose that B = RaU = R3V, where a, BER and U and V are upper- triangular. Prove that if B is invertible, then U = ±V.

(e) Let B be 2 x 2 matrix. any a CO a = Uj1 (i) Show that there are real numbers u11 and a such that sin a a Hint: express as a scalar multiple of a unit vector, and hence find an expression for u11 in terms of a and c. ( ii ) Let α R. Use the invertibility of Ra to prove that there are unique U12, U22 E R such that co a sin a U12 sin a + U22 COs a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = RQU for some a E R and some upper-triangular matrix U. (iv) Suppose that B = RaU = R3V, where a, BER and U and V are upper- triangular. Prove that if B is invertible, then U = ±V.

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

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