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Q-6-5SE

Expert-verifiedFound in: Page 331

Book edition
5th

Author(s)
David C. Lay, Steven R. Lay and Judi J. McDonald

Pages
483 pages

ISBN
978-03219822384

**Show that if an \(n \times n\) matrix satisfies \(\left( {U{\bf{x}}} \right) \cdot \left( {U{\bf{y}}} \right) = {\bf{x}} \cdot {\bf{y}}\) for all x and y in \({\mathbb{R}^n}\), then \(U\) is an orthogonal matrix.**

It is proved that \(U\) is an orthogonal matrix.

**Theorem 7 **states that consider that, \(U\) as an \(m \times n\) matrix with orthonormal columns, and assume that **x** and **y** are in \({\mathbb{R}^n}\). Then;

- \(\left\| {U{\bf{x}}} \right\| = \left\| {\bf{x}} \right\|\)
- \(\left( {U{\bf{x}}} \right) \cdot \left( {U{\bf{y}}} \right) = {\bf{x}} \cdot {\bf{y}}\]
- \(\left( {U{\bf{x}}} \right) \cdot \left( {U{\bf{y}}} \right) = 0\] such that if \({\bf{x}} \cdot {\bf{y}} = 0\).

Assume that, \(\left( {U{\bf{x}}} \right) \cdot \left( {U{\bf{y}}} \right) = {\bf{x}} \cdot {\bf{y}}\) for all \({\bf{x}},{\bf{y}}\) in \({\mathbb{R}^n}\) and consider \({{\mathop{\rm e}\nolimits} _1}, \ldots ,{{\mathop{\rm e}\nolimits} _n}\) as the standard basis for \({\mathbb{R}^n}\).

The \(j{\mathop{\rm th}\nolimits} \) column of \(U\) is denoted by \(U{e_j}\), with \(j = 1, \ldots ,n\). The columns of \(U\) are unit vectors because \({\left\| {U{e_j}} \right\|^2} = \left( {U{e_j}} \right) \cdot \left( {U{e_j}} \right) = {e_j} \cdot {e_j} = 1\).

The columns of \(U\) are pairwise orthogonal because \(\left( {U{e_j}} \right) \cdot \left( {U{e_k}} \right) = {e_j} \cdot {e_k} = 0\).

Thus, it is proved that \(U\) is an orthogonal matrix.

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