Show that the following are equivalent for an n × n matrix P . (a) P is orthogonal. (b) ||Px|| = ||x|| for all columns x e R". (c) ||Px – Py|| = ||x – y|| for all columns x, y e R". (d) (Px) · (Py) = x · y for all columns x, y E R".
Show that the following are equivalent for an n × n matrix P . (a) P is orthogonal. (b) ||Px|| = ||x|| for all columns x e R". (c) ||Px – Py|| = ||x – y|| for all columns x, y e R". (d) (Px) · (Py) = x · y for all columns x, y E R".
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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[Linear Transformations] How do you prove this?
![Show that the following are equivalent for an n x n matrix P .
(a) P is orthogonal.
(b) || Px|| = ||x|| for all columns x e R".
(c) || Px – Py|| = ||x – y||
for all columns x, y e R".
(d) (Px) · (Py) = x·y for all columns x, y e R".](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d1bda18-48b4-4def-b31b-1bfee4373b20%2F0e995768-5ce4-489a-86fe-eb97b858b515%2Fwv8eovi_processed.png&w=3840&q=75)
Transcribed Image Text:Show that the following are equivalent for an n x n matrix P .
(a) P is orthogonal.
(b) || Px|| = ||x|| for all columns x e R".
(c) || Px – Py|| = ||x – y||
for all columns x, y e R".
(d) (Px) · (Py) = x·y for all columns x, y e R".
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