 Suggested languages for you:

Europe

Answers without the blur. Sign up and see all textbooks for free! Q47E

Expert-verified Found in: Page 414 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # 47. If ${\mathbf{A}}{\mathbf{=}}\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right]$, then there exist exactly four orthogonal ${\mathbf{2}}{\mathbf{×}}{\mathbf{2}}$matrices S such that ${{\mathbf{S}}}^{\mathbf{-}\mathbf{1}}$AS is diagonal.

The given statement is FALSE.

See the step by step solution

## Step 1: Check whether the given statement is TRUE or FALSE

Since is symmetric, it has an orthogonal basis consisting of eigenvectors. Let ${\stackrel{˙}{\mathrm{v}}}_{1},{\stackrel{˙}{\mathrm{v}}}_{2}$be unit orthogonal eigenvectors of A. Since S is orthogonal diagonal zing matrix of A,S must map $\left\{{\mathrm{e}}_{1},{\mathrm{e}}_{2}\right\}$to orthogonal eigenvectors.

Thus, we have either ${\mathrm{Se}}_{1}=±{\stackrel{˙}{\mathrm{v}}}_{1}$and ${\mathrm{Se}}_{2}=±{\stackrel{˙}{\mathrm{v}}}_{2}$OR ${\mathrm{Se}}_{1}=±{\stackrel{˙}{\mathrm{v}}}_{2}$ and ${\mathrm{Se}}_{2}=±{\stackrel{˙}{\mathrm{v}}}_{1}$.

This gives a total of 8 possibilities for S.

Therefore, the given statement is FALSE. ### Want to see more solutions like these? 