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]]]