Let \({\bf{x}}\) be an \(m \times n\) real matrix. The
function computes the order \(n\) square matrixmatrix \({\bf{A}} = {\bf{x'}}\;{\bf{x}}\).
The R function eigen is applied to this matrix to obtain the vector
of eigenvalues \({\bf{\lambda }} = \left\lbrack {\begin{array}{cccc}
{\lambda _1 } & {\lambda _2 } & \cdots & {\lambda _n } \\
\end{array}} \right\rbrack\). By construction the eigenvalues are in descending
order of value so that the largest eigenvalue is \(\lambda _1\). Then
the spectral norm is \(\left\| {\bf{x}} \right\|_2 = \sqrt {\lambda _1 }\).
If \({\bf{x}}\) is a vector, then \({\bf{L}}_2 = \sqrt {\bf{A}}\) is returned.
References
Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics,
Society for Industrial and Applied Mathematics.
Golub, G. H. and C. F. Van Loan (1996). Matrix Computations, Third Edition, The John
Hopkins University Press.
Horn, R. A. and C. R. Johnson (1985). Matrix Analysis, Cambridge University Press.