/sci/ - Science & Math

God help me...
Given, [math]\displaystyle T= \begin{bmatrix} 2 & 0 \\
0 & -1
\end{bmatrix}[/math] and [math]X=\mathbb{R}^{2}[/math], I'm trying to find the operator norm, [math]||T||[/math].

By definition, [math]||T|| = \sup\limits_{||x||\neq 0} \frac{ ||Tx|| }{||x||}[/math]. Let [math]x=(a,b)\in \mathbb{R}^{2}[/math]. Then, [math]\displaystyle ||T|| =\sup \frac{\sqrt{4a^{2}+b^{2} } }{\sqrt{a^{2}+b^{2} } } [/math] . Is this the answer? Does it simplify further? Is there an inequality I should be applying? Can I say, let [math]a=b=c>0[/math]. Then [math]\frac{\sqrt{4c^{2}+c^{2} } }{\sqrt{c^{2}+c^{2} } } \leq \frac{\sqrt{5c^{2}}}{\sqrt{2c^{2}}} = c\sqrt{\frac{5}{2}}[/math] ?

>>10090833
>>10090835
>>10090843
>>10090858
>>10090894

>>10060336
bump