I have a linear operator $T:\mathbb{R}^3\rightarrow \mathbb{R}^2$ defined by $T\begin{bmatrix}x\\y\\z \end{bmatrix}=\begin{bmatrix}1&0&0\\0&1&1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}.$ The domain and codomain spaces are having Euclidean norms on them. Then what is the norm of $T.$ The definition for a norm here is $\|T\|=sup \Big\{\frac{\|Tv\|_2}{\|v\|_2} \hspace{0.05 cm}:v\neq 0\Big\}$
How to compute the operator norm for the following linear operator?
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matrices
norm
linear-transformations
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0It is $\sqrt2$. – 2017-02-16
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0How to compute? – 2017-02-16
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0First you need the correct definition, it is a $\sup$ rather than $\inf$. Then there are many possible methods, and which you might want to use depends on context that is absent. Maybe just try computing $\|Tv\|/\|v\|$ in generic form and see how you can bound it? – 2017-02-16
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0Yes you are right.. its supremum. Then if it is supremum how to calculate? – 2017-02-16
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0Do you mean it is 2-norm of a matrix i.e $\|T\|=\sqrt{a}$ where $a=$maximum eigen value of $T^*T$ – 2017-02-16
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0That is true, but I don't know if I meant it. In this case TT* is easier. – 2017-02-16