matrix p-norm (original) (raw)
| ∥A∥p=supx≠0∥Ax∥p∥x∥p x∈ℝn,A∈ℝm×n. |
|---|
The matrix p-norms are defined in terms of the vector p-norms (http://planetmath.org/VectorPNorm).
As with vector p-norms, the most important are the 1, 2, and ∞ norms. The 1 and ∞ norms are very easy to calculate for an arbitrary matrix:
| ∥A∥1=max1≤j≤n∑i=1m|aij|∥A∥∞=max1≤i≤m∑j=1n|aij|. | | -------------------------------------------------------- |
It directly follows from this that ∥A∥1=∥AT∥∞.
The calculation of the 2-norm is more complicated. However, it can be shown that the 2-norm of A is the square root of the largest eigenvalue
of ATA. There are also various inequalities
that allow one to make estimates on the value of ∥A∥2: