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Q-6-5SE

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Linear Algebra and its Applications
Found in: Page 331
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

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.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Statement in Theorem 7

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;

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

Show that \(U\] is an orthogonal matrix

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.

Most popular questions for Math Textbooks

Use the Gram–Schmidt process as in Example 2 to produce an orthogonal basis for the column space of

\(A = \left( {\begin{aligned}{{}{r}}{ - 10}&{13}&7&{ - 11}\\2&1&{ - 5}&3\\{ - 6}&3&{13}&{ - 3}\\{16}&{ - 16}&{ - 2}&5\\2&1&{ - 5}&{ - 7}\end{aligned}} \right)\)

In exercises 1-6, determine which sets of vectors are orthogonal.

\(\left[ {\begin{align}{ 2}\\{ - 7}\\{-1}\end{align}} \right]\), \(\left[ {\begin{align}{ - 6}\\{ - 3}\\9\end{align}} \right]\), \(\left[ {\begin{align}{ 3}\\{ 1}\\{-1}\end{align}} \right]\)

In Exercises 7–10, let \[W\] be the subspace spanned by the \[{\bf{u}}\]’s, and write y as the sum of a vector in \[W\] and a vector orthogonal to \[W\] .

8. \[y = \left[ {\begin{aligned}{ - 1}\\4\\3\end{aligned}} \right]\], \[{{\bf{u}}_1} = \left[ {\begin{aligned}1\\1\\{\bf{1}}\end{aligned}} \right]\], \[{{\bf{u}}_2} = \left[ {\begin{aligned}{ - 1}\\3\\{ - 2}\end{aligned}} \right]\]

Show that if \(U\) is an orthogonal matrix, then any real eigenvalue of \(U\) must be \( \pm 1\).

In Exercises 13 and 14, the columns of Q were obtained by applying the Gram-Schmidt process to the columns of A. Find an upper triangular matrix R such that \(A = QR\). Check your work.

13. \(A = \left( {\begin{aligned}{{}{}}5&9\\1&7\\{ - 3}&{ - 5}\\1&5\end{aligned}} \right),{\rm{ }}Q = \left( {\begin{aligned}{{}{}}{\frac{5}{6}}&{ - \frac{1}{6}}\\{\frac{1}{6}}&{\frac{5}{6}}\\{ - \frac{3}{6}}&{\frac{1}{6}}\\{\frac{1}{6}}&{\frac{3}{6}}\end{aligned}} \right)\)
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