matrixcalc (version 1.0-3)

spectral.norm: Spectral norm of matrix

Description

This function returns the spectral norm of a real matrix.

Usage

spectral.norm(x)

Arguments

x

a numeric matrix or vector

Value

A numeric value.

Details

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 \({\bf{\lambda }} = \left[ {\begin{array}{*{20}c} {\lambda _1 } & {\lambda _2 } & \cdots & {\lambda _n } \\ \end{array}} \right]\). 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.

Examples

# NOT RUN {
x <- matrix( c( 2, 4, 2, 1, 3, 1, 5, 2, 1, 2, 3, 3 ), nrow=4, ncol=4, byrow=TRUE )
spectral.norm( x )
# }