Operator Norm

The operator norm of a linear operator is the largest value by which stretches an element of ,

(1)

It is necessary for and to be normed vector spaces. The operator norm of a composition is controlled by the norms of the operators,

(2)

When is given by a matrix, say , then is the square root of the largest eigenvalue of the symmetric matrix , all of whose eigenvalues are nonnegative. For instance, if

(3)

then

(4)

which has eigenvalues , so .

The following Wolfram Language code will determine the operator norm of a matrix:

  OperatorNorm[a_List?MatrixQ] :=
    Sqrt[Max[Eigenvalues[Transpose[a].a]]]