Coordinate norm is a kind of vector norm, also called p norm. It is to sum the absolute values of each coordinate of a vector to the power of p, and then find its root of p, that is, |||||||| p = (| x1| p +| x2 | p+... +| xn | p) (65438).

N is the dimension of the vector. For example, when p= 1, the coordinate norm is the sum of the absolute values of the coordinates in each dimension of the vector, and when p=2, it is the Euclidean norm of the vector, indicating the length of the vector.

Norm is a basic concept in mathematics. In functional analysis, it is defined in a normed linear space and satisfies certain conditions, that is, nonnegativity; Homogeneity; Trigonometric inequality is often used to measure the length or size of each vector in a vector space (or matrix).

Nominal definition

Norm is a function with the concept of "length". In the fields of linear algebra, functional analysis and related mathematics, norm is a function that gives all vectors in vector space non-zero positive length or size. A semi-norm can give a nonzero vector a zero length.

The vector space defining the norm is a normed vector space; Similarly, the vector space that defines a seminorm is a seminormed vector space.

Matrix norm

If║║║is a compatible specification, and any norm║║║║that satisfies║║║is is not a compatible specification, then║║║is called it. For any norm ║║║║║║║║║║║║║║║║║║║║║║║║ιιιιιι 0, so k ║ ║ is the minimum norm.

Note: If compatibility is not considered, there is no difference between matrix norm and vector norm, because all mxn matrices are isomorphic with mn-dimensional vector space. The main purpose of introducing compatibility is to keep the characteristics of matrix as a linear operator, which is consistent with the compatibility of operator norm, and can get information other than Mincowski theorem.