Spectral norm of matrix
This function returns the spectral norm of a real matrix.
spectral.norm(x)
x |
a numeric matrix or vector |
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{λ }} = ≤ft[ {\begin{array}{*{20}c}
{λ _1 } & {λ _2 } & \cdots & {λ _n } \\
\end{array}} \right]. By construction the eigenvalues are in descending
order of value so that the largest eigenvalue is λ _1. Then
the spectral norm is ≤ft\| {\bf{x}} \right\|_2 = √ {λ _1 }.
If {\bf{x}} is a vector, then {\bf{L}}_2 = √ {\bf{A}} is returned.
A numeric value.
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.
Frederick Novomestky fnovomes@poly.edu
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.
x <- matrix( c( 2, 4, 2, 1, 3, 1, 5, 2, 1, 2, 3, 3 ), nrow=4, ncol=4, byrow=TRUE ) spectral.norm( x )
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