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Q59E

Expert-verifiedFound in: Page 109

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**TRUE OR FALSE?**

**If a matrix **${\mathit{A}}{\mathbf{=}}{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)}$ **represents the orthogonal projection onto a line L** **, then the equation **${\mathit{a}}^{\mathbf{2}}\mathbf{+}{\mathit{b}}^{\mathbf{2}}\mathbf{+}{\mathit{c}}^{\mathbf{2}}\mathbf{+}{\mathit{d}}^{\mathbf{2}}\mathbf{=}{\mathbf{1}}$ **must hold. **

The given statement is false.

The orthogonal projection of matrix is of form $\left(\begin{array}{cc}{u}_{1}^{2}& {u}_{1}{u}_{2}\\ {u}_{1}{u}_{2}& {u}_{2}^{2}\end{array}\right)$ with $\overrightarrow{u}=\left[\begin{array}{c}{u}_{1}\\ {u}_{2}\end{array}\right]$. Thus, if

${\mathrm{a}}^{2}+{\mathrm{b}}^{2}+{\mathrm{c}}^{2}+{\mathrm{d}}^{2}=1\mathrm{then}$

${\begin{array}{cc}{u}_{1}^{4}+& \left({u}_{1}{u}_{2}\right)\end{array}}^{2}+{\left({u}_{1}{u}_{2}\right)}^{2}+{u}_{2}^{4}=1\phantom{\rule{0ex}{0ex}}\Rightarrow {\left({u}_{1}{u}_{2}\right)}^{2}=1$

Consider the matrix.

$A=\left(\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right)$

If $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ then,

The elements will be,

$a={u}_{1}^{2},d={u}_{2}^{2},b=c={u}_{1}{u}_{2}$

Compute, ${a}^{2}+{b}^{2}+{c}^{2}+{d}^{2}$

${a}^{2}+{b}^{2}+{c}^{2}+{d}^{2}={\left({u}_{1}^{2}\right)}^{2}+{\left({u}_{1}{u}_{2}\right)}^{2}+{\left({u}_{1}{u}_{2}\right)}^{2}+{\left({u}_{2}^{2}\right)}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow {a}^{2}+{b}^{2}+{c}^{2}+{d}^{2}=\left({u}_{1}^{2}+{u}_{2}^{2}\right)\left({u}_{1}^{2}+{u}_{2}^{2}\right)$

Hence, the statement is false.

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