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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
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".
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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