Let
$$A = \begin{pmatrix}
-3 & -4 & -2 \\
5 & 9 & -5 \\
-3 & 8 & -9
\end{pmatrix}$$Calculate the following norms of matrix $A$.
a. $\|A\|_{1,1}$
b. $\|A\|_{∞,∞}$
c. $\|A\|_{1,∞}$
d. $\|A\|_{2,∞}$
e. $\|A\|_{1,2}$
I know that $||A||_1=21$ and $||A||_\infty=20$, but I'm not sure what to do after that for option a or option b. I haven't done $||A||_2$ yet, but I know how to get that information. I just don't know what do when I have $(1,1)$, but especially letters $c, d, $ and $e$. Any help would be appreciated.
I have the following Lemma:
$$ \textrm{ Lemma 7.22. If } A \in \mathbb{C}^{p \times q} \textrm{ then } $$
$$ \textrm{1.} \| A\|_{1,1} = \max_{1 \leq j \leq q} \Big\{ \sum_{i=1}^{p} |a_{ij}|\Big\} \textrm{ ( maximum column sum ( modulus))}$$
$$ \textrm{2.} \| A\|_{\infty,\infty} = \max_{1 \leq i \leq p} \Big\{ \sum_{j=1}^{q} |a_{ij}|\Big\} \textrm{ ( maximum row sum (modulus) )}$$
$$ \textrm{3.} \| A\|_{2,2} = s_{1} \textrm{ where } s_{1}^{2} \textrm{ is the maximum eigenvalue of the matrix } A^{H}A $$
$$ \textrm{4.} \| A\|_{1,\infty} = \max_{i,j} |a_{ij}| \textrm{ ( maximum (modulus) )} $$
$$ \textrm{5.} \| A\|_{2,\infty} = \max_{1 \leq i \leq p} \Big\{ \bigg( \sum_{j=1}^{q} |a_{ij}|^{2} \bigg)^{\frac{1}{2}} \Big\} \textrm{ ( maximum 2-norm of rows)}$$
$$ \textrm{6.} \| A\|_{1,2} = \max_{j} \Big\{ \bigg( \sum_{i=1}^{p} |a_{ij}|^{2} \bigg)^{\frac{1}{2}} \Big\} \textrm{ ( maximum 2-norm of columns)}$$
Best Answer
If you have your matrix $A$ given as the following
$$A = \begin{pmatrix} -3 & -4 & -2 \\ 5 & 9 & -5 \\ -3 & 8 & -9\end{pmatrix} $$
then we have
$$ \textrm{a.} \| A\|_{1,1} = \max_{1 \leq j \leq q} \Big\{ \sum_{i=1}^{p} |a_{ij}|\Big\} \textrm{ ( maximum column sum ( modulus))}$$ This is the maximum sum of the absolute values of the columns
$$ \sum_{i=1}^{p} |a_{i1} | = 3 + 5 + 3 = 11 \\ \sum_{i=1}^{p} |a_{i2}| = 4 + 9 + 8 = 21 \\ \sum_{i=1}^{p} |a_{i3}| = 2 + 5+ 9 = 16 $$
which gives us $\|A\|_{1,1} = 21$
$$ \textrm{b.} \| A\|_{\infty,\infty} = \max_{1 \leq i \leq p} \Big\{ \sum_{j=1}^{q} |a_{ij}|\Big\} \textrm{ ( maximum row sum (modulus) )}$$
which is the maximum row sum $$ \sum_{j=1}^{q} |a_{1j}| = 3 + 4 + 2 = 9 \\ \sum_{j=1}^{q} |a_{2j}| = 5 + 9 + 5 = 19 \\ \sum_{j=1}^{q} |a_{3j}| = 3+8+9 = 20$$
$$ \|A\|_{\infty, \infty} = 20$$
$$ \textrm{c.} \| A \|_{1,\infty} = \max_{i,j} |a_{ij}| $$
which is the largest absolute value of some entry, $a_{22}$ and $a_{33} $ have absolute value $9$
$$ \| A \|_{1,\infty} = 9 $$
$$ \textrm{d.} \| A\|_{2,\infty} = \max_{1 \leq i \leq p} \Big\{ \bigg( \sum_{j=1}^{q} |a_{ij}|^{2} \bigg)^{\frac{1}{2}} \Big\} \textrm{ ( maximum 2-norm of rows)}$$
which is the maximum $2$ norm of the rows
$$ \bigg(\sum_{j=1}^{q} |a_{1j}|^{2} \bigg)^{\frac{1}{2}} = \sqrt{ 3^{2} + 4^{2} + 2^{2} } = \sqrt{17} $$
$$ \bigg(\sum_{j=1}^{q} |a_{2j}|^{2} \bigg)^{\frac{1}{2}} = \sqrt{ 5^{2} + 9^{2} + 5^{2} } = \sqrt{131} $$
$$ \bigg(\sum_{j=1}^{q} |a_{3j}|^{2} \bigg)^{\frac{1}{2}} = \sqrt{ 3^{2} + 8^{2} + 9^{2} } = \sqrt{154} $$
we see that $\|A\|_{2,\infty} = \sqrt{154}$
Finally for the last one
$$ \textrm{e.} \| A\|_{1,2} = \max_{j} \Big\{ \bigg( \sum_{i=1}^{p} |a_{ij}|^{2} \bigg)^{\frac{1}{2}} \Big\} \textrm{ ( maximum 2-norm of columns)}$$
is the maximum $2$ norm of the columns
$$ \bigg(\sum_{j=1}^{q} |a_{i1}|^{2} \bigg)^{\frac{1}{2}} = \sqrt{3^{2}+ 5^{2} + 3^{2}} = \sqrt{43} $$
$$ \bigg(\sum_{j=1}^{q} |a_{i2}|^{2} \bigg)^{\frac{1}{2}} = \sqrt{4^{2}+ 9^{2} + 8^{2}} = \sqrt{161} $$
$$ \bigg(\sum_{j=1}^{q} |a_{i3}|^{2} \bigg)^{\frac{1}{2}} = \sqrt{2^{2}+ 5^{2} + 9^{2}} = \sqrt{110} $$
$$ \|A \|_{1,2} = \sqrt{161} $$