91Ó°ÊÓ

Given that,

$A=\left[\begin{array}{cc}1& 0\\ 0& -2\end{array}\right]$uncaught exception: Http Error #403

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #403') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Http Error #403') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #403') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Http Error #403') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #403') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('98d397235709778...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('

Now the singular values of A are found by first computing the eigenvalues of the square matrix

Then:

Let characteristic polynomial ofbe p(x). Then:

Thus the characteristic polynomial isThis implies that the roots of p(x) are 1,4 . Hence the eigenvalues ofareandThus the singular values of A areand

Now let

X₁ = x₁

x₂

be an eigenvector corresponding to the eigenvalue 4. Therefore (A^tA)X₁=4X₁which means (4I₂-A^tA)X₁=0. From this equation we get that

Therefore

3x₁= 0

This implies that x₁= 0.

Therefore is an eigenvector corresponding to the eigenvalue 4 for any non-zero real number r. By putting r=1, we conclude that is an eigenvector corresponding to the eigenvalue 4. Thus, is a unit eigenvector of corresponding to the eigenvalue

Now let

be an eigenvector corresponding to the eigenvalue 1 . Thereforewhich meansFrom this equation we get that

Therefore,.

This implies that

Therefore.

is an eigenvector corresponding to the eigenvalue 1 for any non-zero real number r. By putting r=1, we conclude that.

is an eigenvector corresponding to the eigenvalue 1. Thusis a unit eigenvector ofcorresponding to the eigenvalue

Hence we have:

The singular values of A areand

Thus

Now to find U, we compute the columns and of U as follows:

Thus

Therefore, a singular value decomposition of A is then given by

The image of the unit circle has the following form:

A singular value decomposition of A is given by $A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$uncaught exception: Http Error #403

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #403') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Http Error #403') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #403') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Http Error #403') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #403') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('0b6089141dda374...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('" width="9" height="19" role="math">" width="9" height="19" role="math">