Operator Norm

The operator norm of a bounded operator between normed spaces is the supremum of the factors by which stretches a nonzero element of . For ,

(1)

When , the operator norm is 0. For bounded operators and , with also a normed space, the operator norm of their composition is controlled by their norms,

(2)

When is given by an real matrix or complex matrix with , say , and both spaces have their Euclidean norms, is the square root of the largest eigenvalue of the Hermitian matrix , where is the conjugate transpose. These eigenvalues are nonnegative. For a real matrix, . For instance, if

(3)

then

(4)

which has eigenvalues , so .

The following Wolfram Language code will determine the Euclidean operator norm of a real matrix or complex matrix:

OperatorNorm[a_List?MatrixQ] :=
 Norm[a, 2]