Suppose the invertible n x n matrices U and V satisfy the special properties U2 V2 and UVU = V. (a) Use the special properties of U and V above to prove that Umvum - V, for all positive integers m. Essay box advice: In the essay boxes below, there is no need to use correct Maple syntax or use the equation editor, as long as what you write is clear for a reader. For example, . UV may be written as 'UV', . Um may be written as 'U^m', U-1 . may be written as 'U^(-1), The identity matrix I may be written as 'T'.
Suppose the invertible n x n matrices U and V satisfy the special properties U2 V2 and UVU = V. (a) Use the special properties of U and V above to prove that Umvum - V, for all positive integers m. Essay box advice: In the essay boxes below, there is no need to use correct Maple syntax or use the equation editor, as long as what you write is clear for a reader. For example, . UV may be written as 'UV', . Um may be written as 'U^m', U-1 . may be written as 'U^(-1), The identity matrix I may be written as 'T'.
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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Question
Transcribed Image Text:Suppose the invertible n x n matrices U and V satisfy the special properties
U2 V2 and UVU = V.
(a) Use the special properties of U and V above to prove that UmVU = V, for all positive integers m.
Essay box advice: In the essay boxes below, there is no need to use correct Maple syntax or use the equation
editor, as long as what you write is clear for a reader.
For example,
.
UV may be written as 'UV',
. Um may be written as 'U^m',
.
U-1 may be written as 'U^(-1),
. The identity matrix I may be written as 'T.
![(b) Next, suppose we construct the matrices
X = UPV9 and Y = Ujvk
where p, q, j, k are positive integers, and q is odd. Using part (a) or otherwise, one can show that
XY = Uav
where a and B are indices which may be given in terms of p, q, j, k by [Select all that apply]
Multiple selection advice: In a multiple selection question, marks are deducted for incorrect selections (but you
cannot get less than zero for it). You are advised to only select options that you are sure about.
□a=p-j-1 and ß = q + k + 1
□a=p+j+3 and B=q+k+2
4
□a=p-jand B = q + k
□a=p+j+1 and ß = q + k
a=p-j-2 and B=q+k+2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ece5e93-8b11-4de2-b105-809b648d3f91%2Fe934d687-b3f9-4986-a901-13bdb022655a%2Ftxodf5c_processed.png&w=3840&q=75)
Transcribed Image Text:(b) Next, suppose we construct the matrices
X = UPV9 and Y = Ujvk
where p, q, j, k are positive integers, and q is odd. Using part (a) or otherwise, one can show that
XY = Uav
where a and B are indices which may be given in terms of p, q, j, k by [Select all that apply]
Multiple selection advice: In a multiple selection question, marks are deducted for incorrect selections (but you
cannot get less than zero for it). You are advised to only select options that you are sure about.
□a=p-j-1 and ß = q + k + 1
□a=p+j+3 and B=q+k+2
4
□a=p-jand B = q + k
□a=p+j+1 and ß = q + k
a=p-j-2 and B=q+k+2
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