Suggested languages for you:

Americas

Europe

Q47E

Expert-verifiedFound in: Page 414

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 ${\mathbf{2}}{\mathbf{\times}}{\mathbf{2}}$****matrices S** **such that ${{\mathbf{S}}}^{\mathbf{-}\mathbf{1}}$AS**** is diagonal.**

The given statement is FALSE.

Since is symmetric, it has an orthogonal basis consisting of eigenvectors. Let ${\dot{\mathrm{v}}}_{1},{\dot{\mathrm{v}}}_{2}$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 ${\mathrm{Se}}_{1}=\pm {\dot{\mathrm{v}}}_{1}$and ${\mathrm{Se}}_{2}=\pm {\dot{\mathrm{v}}}_{2}$OR ${\mathrm{Se}}_{1}=\pm {\dot{\mathrm{v}}}_{2}$ and ${\mathrm{Se}}_{2}=\pm {\dot{\mathrm{v}}}_{1}$.

This gives a total of 8 possibilities for S.

Therefore, the given statement is FALSE.

94% of StudySmarter users get better grades.

Sign up for free