Editing Matrix norm (section)

===Matrix norms induced by vector ''α''- and ''β''-norms===
We can generalize the above definition. Suppose we have vector norms <math>\|\cdot\|_{\alpha}</math> and <math>\|\cdot\|_{\beta}</math> for spaces <math>K^n</math> and <math>K^m</math> respectively; the corresponding operator norm is
<math display="block">
\|A\|_{\alpha, \beta} = \sup\{ \|Ax\|_\beta : x \in K^n \text{ such that } \|x\|_\alpha \leq 1 \}
</math>
In particular, the <math>\|A\|_{p}</math> defined previously is the special case of <math>\|A\|_{p, p}</math>.

In the special cases of <math>\alpha = 2</math> and <math>\beta=\infty</math>, the induced matrix norms can be computed by<math display="block"> \|A\|_{2,\infty}= \max_{1\le i\le m}\|A_{i:}\|_2, </math> where <math>A_{i:}</math> is the i-th row of matrix <math> A </math>.

In the special cases of <math>\alpha = 1</math> and <math>\beta=2</math>, the induced matrix norms can be computed by<math display="block"> \|A\|_{1, 2} = \max_{1\le j\le n}\|A_{:j}\|_2, </math> where <math>A_{:j}</math> is the j-th column of matrix <math> A </math>.

Hence, <math> \|A\|_{2,\infty} </math> and <math> \|A\|_{1, 2} </math> are the maximum row and column 2-norm of the matrix, respectively.