a simple question about a matrix

There are many matrix norms.  The Mathematica web site 
(http://mathworld.wolfram.com/MatrixNorm.html) says the one Andy just 
gave 'is often referred to as "the" matrix norm.'  However, there many 
others.  For some purposes, any matrix norm will do.  For other 
purposes, you need to know the purpose. 

      Please excuse me if this seems like a picky, pedantic rant.  
spencer graves

If the matrix is real, then isn't that just
sqrt(eigen(crossprod(x))$values[1]) ??

Andy

 

From: li xian

suppose x is a matrix.
how to define ||x|| in R?

thank you!


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"Liaw, Andy" <andy_liaw at merck.com> writes:

If the matrix is real, then isn't that just
sqrt(eigen(crossprod(x))$values[1]) ??

Andy

From: li xian

suppose x is a matrix.
how to define ||x|| in R?
 

The symbol ||x|| could mean many different things when x is a matrix.
I presume that you mean some kind of matrix norm.  Could you be more
specific about which norm you want?  (There are several.)

The one-norm, the Frobenius norm, and the infinity norm are all
available in the new version of the Matrix package, which is currently
available only for the development version of R (to be R-1.9.0).

mm = Matrix(rnorm(9), nrow = 3)
mm

[,1]       [,2]      [,3]
[1,] -1.0210873  0.9941221  2.363475
[2,] -0.5171302 -0.3243187 -1.455873
[3,] -2.3793565  0.5223258 -1.034270

norm(mm, '1')

[1] 4.853617

norm(mm, 'O')   # the one norm

[1] 4.853617

norm(mm, 'I')   # the infinity norm

[1] 4.378684

norm(mm, 'F')   # the Frobenius norm

[1] 4.136781

norm(mm, 'M')   # the Maximum modulus (not really a norm)

[1] 2.379356