Induced norm

现在的位置: 首页 > 综合 > 正文

2013年01月24日 ⁄ 综合 ⁄ 共 1165字 ⁄ 字号小中大 ⁄ 评论关闭

Induced
norm

If vector norms on K^m and Kⁿ are
given (K is field of real or complex
numbers), then one defines the corresponding induced norm or operator norm on the space
of m-by-n matrices as the following maxima:

$\begin{align} \|A\| &= \max\{\|Ax\| <wbr>: x\in K^n \mbox{ with }\|x\|\le 1\} \ &= \max\{\|Ax\| <wbr>: x\in K^n \mbox{ with }\|x\| = 1\} \ &= \max\left\{\frac{\|Ax\|}{\|x\|} <wbr>: x\in K^n \mbox{ with }x\ne 0\right\}. \end{align}$

These are different from the entrywise p-norms and the Schatten p-norms for matrices treated below, which are also usually denoted by $\left \| A \right \| _p .$

If m = n and one uses the same norm on the domain and the range, then the induced operator norm is a sub-multiplicative matrix norm.

The operator norm corresponding to the p-norm for vectors is:

$\left \| A \right \| _p = \max \limits _{x \ne 0} \frac{\left \| A x\right \| _p}{\left \| x\right \| _p}.$

In the case of p = 1 and $p=\infty$ ,
the norms can be computed as:

$\left \| A \right \| _1 = \max \limits _{1 \leq j \leq n} \sum _{i=1} ^m | a_{ij} |,$ which
is simply the maximum absolute column sum of the matrix

$\left \| A \right \| _\infty = \max \limits _{1 \leq i \leq m} \sum _{j=1} ^n | a_{ij} |,$ which
is simply the maximum absolute row sum of the matrix

For example, if the matrix A is defined by

$A = \begin{bmatrix} 3 & 5 & 7 \ 2 & -6 & 4 \ 0 & 2 & 8 \ \end{bmatrix},$

then we have ||A||₁ = 7+4+8 = 19. and ||A||_∞ =
3+5+7 = 15

In the special case of p = 2 (the Euclidean norm)
and m = n (square matrices), the induced matrix norm is the spectral norm. The spectral norm of a matrix A is the largest singular
value of A or the square root of the largest eigenvalue of the positive-semidefinite
matrix A^*A: