The overall norm of a real or complex ( m × n ) matrix with as the field of real or complex numbers is defined as
,
thus the product of the geometric mean of the number of rows and columns in the matrix with the maximum of the amounts of all matrix elements . With the exception of the prefactor, the overall standard thus corresponds to the maximum entry of a vector of length in which all entries in the matrix are noted below one another, and thus to the maximum standard of this vector.
For the special case of a square matrix , the overall norm is through
given.
Examples
Real matrix
The overall norm of the real (2 × 2) matrix
is given as
.
Complex matrix
The overall norm of the complex (2 × 2) matrix
is given as
.
properties
Norm axioms
Since the sum of two matrices and the multiplication of a matrix by a scalar are defined component-wise, the norm properties of definiteness , absolute homogeneity and subadditivity follow directly from the corresponding properties of the maximum norm for vectors. The scaling with the constant prefactor has no influence on the statements.
like with the help of the triangle inequality and with the estimation of a sum of matrix elements by the corresponding multiple of the maximum element over
can be shown. This also explains the reason for the scaling, since the overall standard is generally not sub-multiplicative without this prefactor.
compatibility
The overall standard is compatible with all p standards , provided that applies to and to . Under these restrictions, the inequality
holds for a matrix and a vector
.
The compatibility follows from the chain of inequalities
where the prefactor is bounded by one under exactly the above conditions . The 1-norm was estimated by the p -norm and, as with submultiplicativity, the sum was replaced by the maximum and the triangular inequality was repeatedly applied.
The overall norm is therefore always compatible with the Euclidean norm . It is only compatible with the sum norm and all other p norms for if the number of rows is at most as large as that of the columns. It is only compatible with the maximum norm and all other p norms for if the number of rows is at least as large as that of columns. For square matrices the overall norm is compatible with all p norms.
applies, as each operator norm for the identity matrix the value one must have, but for greater results in a value less than one. If the overall norm for the identity matrix is scaled to one, then the submultiplicativity, which is another property of every operator norm, is lost.