matrix norm

How much a matrix can stretch a vector, at most -- extending a vector norm's own 'how big is this' question one level up, to 'how much can this matrix amplify any vector's size.'

A matrix norm induced by a vector norm is $\lVert A\rVert=\max_{\lVert\mathbf{x}\rVert=1}\lVert A\mathbf{x}\rVert$ , the largest factor by which $A$ can stretch a unit vector. The infinity-norm ( $\max_i\sum_j|a_{ij}|$ , the largest absolute row sum) and one-norm ( $\max_j\sum_i|a_{ij}|$ , the largest absolute column sum) are hand-computable directly from $A$ 's entries; the two-norm is geometrically the image ellipse's longest semi-axis, but needs a singular value to pin down exactly.

Examples

For $A=\begin{pmatrix}1&-2\\3&4\end{pmatrix}$ : row sums $|1|+|-2|=3$ and $|3|+|4|=7$ give $\lVert A\rVert_\infty=7$ .