16. (a) Prove that if Q is an orthogonal matrix, then ||Qx|| = ||x || for every vector x ER" where ||· || denotes the standard Euclidean norm. (b) Prove the converse: if ||Qx|| = ||×|| for all x € R", then Q is an orthogonal matrix.
16. (a) Prove that if Q is an orthogonal matrix, then ||Qx|| = ||x || for every vector x ER" where ||· || denotes the standard Euclidean norm. (b) Prove the converse: if ||Qx|| = ||×|| for all x € R", then Q is an orthogonal matrix.
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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![4.3.16. (a) Prove that if Q is an orthogonal matrix, then ||Qx|| = ||x || for every vector x = R",
where || || denotes the standard Euclidean norm. (b) Prove the converse: if ||Qx|| = ||x||
for all x ER", then Q is an orthogonal matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc31065e4-6071-4dbc-99a3-4d8b8e7c0b94%2F15847404-405d-4b58-b546-6d2be42cd7f5%2Ff94a6ba_processed.png&w=3840&q=75)
Transcribed Image Text:4.3.16. (a) Prove that if Q is an orthogonal matrix, then ||Qx|| = ||x || for every vector x = R",
where || || denotes the standard Euclidean norm. (b) Prove the converse: if ||Qx|| = ||x||
for all x ER", then Q is an orthogonal matrix.
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