| spectral.norm {matrixcalc} | R Documentation |
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 |
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\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.
Value
A numeric value.
Note
If the argument x is not numeric, an error message is displayed and the function terminates.
If the argument is neither a matrix nor a vector, an error message is displayed and the
function terminates.
If the product matrix {\bf{x'}}\;{\bf{x}} is negative definite, an error message
displayed and the function terminates.
Author(s)
Frederick Novomestky fnovomes@poly.edu
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
x <- matrix( c( 2, 4, 2, 1, 3, 1, 5, 2, 1, 2, 3, 3 ), nrow=3, ncol=4, byrow=TRUE )
spectral.norm( x )