(Matrices) From 2.7-7 we know that an r ×n matrix A=(αj k) defines a linear operator from the vector space X of all ordered n-tuples of numbers into the vector space Y of all ordered r-tuples of numbers. Suppose that any norm ·1 is given on X and any norm ·2 is given on Y. Remember from Prob. 10, Sec. 2.4, that there are various norms on the space Z of all those matrices ( r and n fixed). A norm · on Z is said to be compatible with ·1 and ·2 if A x2 ≦Ax1 . Show that the norm defined by A=supx ∈x x ≠0 (A x2)/(x1) is compatible with ·1 and ·2. This norm is often called the natural norm defined by ·1 and ·2 . If we choose x1=maxt|ξ| and y2=max1|ηk|, show that the natural norm is A=maxi ∑k=1^n|αj k|

(Matrices) From 2.7-7 we know that an $r \times n$ matrix $A=\left(\alpha_{j k}\right)$ defines a linear operator from the vector space $X$ of all ordered $n$-tuples of numbers into the vector space $Y$ of all ordered $r$-tuples of numbers. Suppose that any norm $\|\cdot\|_{1}$ is given on $X$ and any norm $\|\cdot\|_{2}$ is given on $Y$. Remember from Prob. $10, \mathrm{Sec}$. 2.4, that there are various norms on the space $Z$ of all those matrices ( $r$ and $n$ fixed). A norm $\|\cdot\|$ on $Z$ is said to be compatible with $\|\cdot\|_{1}$ and $\|\cdot\|_{2}$ if $$ \|A x\|_{2} \leqq\|A\|\|x\|_{1} . $$ Show that the norm defined by $$ \|\boldsymbol{A}\|=\sup _{x \in x \atop x \neq 0} \frac{\|A x\|_{2}}{\|x\|_{1}} $$ is compatible with $\|\cdot\|_{1}$ and $\|\cdot\|_{2}$. This norm is often called the natural norm defined by $\|\cdot\|_{1}$ and $\|\cdot\|_{2} .$ If we choose $\|x\|_{1}=\max _{t}|\xi|$ and $\|y\|_{2}=\max _{1}\left|\eta_{k}\right|$, show that the natural norm is $$ \|A\|=\max _{i} \sum_{k=1}^{n}\left|\alpha_{j k}\right| $$

(Matrices) From 2.7-7 we know that an $r \times n$ matrix $A=\left(\alpha_{j k}\right)$ defines a linear operator from the vector space $X$ of all ordered $n$-tuples of numbers into the vector space $Y$ of all ordered $r$-tuples of numbers. Suppose that any norm $\|\cdot\|_{1}$ is given on $X$ and any norm $\|\cdot\|_{2}$ is given on $Y$. Remember from Prob. $10, \mathrm{Sec}$. 2.4, that there are various norms on the space $Z$ of all those matrices ( $r$ and $n$ fixed). A norm $\|\cdot\|$ on $Z$ is said to be compatible with $\|\cdot\|_{1}$ and $\|\cdot\|_{2}$ if
$$
\|A x\|_{2} \leqq\|A\|\|x\|_{1} .
$$
Show that the norm defined by
$$
\|\boldsymbol{A}\|=\sup _{x \in x \atop x \neq 0} \frac{\|A x\|_{2}}{\|x\|_{1}}
$$
is compatible with $\|\cdot\|_{1}$ and $\|\cdot\|_{2}$. This norm is often called the natural norm defined by $\|\cdot\|_{1}$ and $\|\cdot\|_{2} .$ If we choose $\|x\|_{1}=\max _{t}|\xi|$ and $\|y\|_{2}=\max _{1}\left|\eta_{k}\right|$, show that the natural norm is
$$
\|A\|=\max _{i} \sum_{k=1}^{n}\left|\alpha_{j k}\right|
$$

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