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Generalized Eigenvector and Singular Value Decompositions

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  • Subject: [mg304] Generalized Eigenvector and Singular Value Decompositions
  • From: Mark Kotanchek <mek at>
  • Date: Wed, 7 Dec 94 19:51:04 -0500


In my research, I need to find the generalized singular values of the matrix  

	| A.x - e B.x | = 0

where A and B are possibly non-square complex-valued matrices and "e" is the  
singular values. My problem is that Mathematica has neither the generalized  
singular value nor the generalized eigenvalue decomposition in its  

I posed a query to WRI about my needs:

   In my research in array signal processing, I often need to
   resort to linear   algebra -- most notably eigendecompositions
   (EVDs), and singular value   decompositions (SVDs).
   Unfortunately, I have a few problems with Mma:

   1) the SVD of a 6x6 complex-valued matrix requires 0.8 seconds
   on my   NeXTstation -- which is IMHO excessive;

   2) there is no generalized eigendecomposition routine to
   solve equations of   the form, A.x - yB.x = 0 where A and B are square

   3) there is not generalized singular value decomposition to
   solve equations   of the form, |A - yB| = 0 where A and B are
   non-square matrices; and,

   4) the pseudo-inverse function appears to not be very robust
   and occasionally   fails against non-square matrices which
   should not pose a problem.

   Since these are fairly fundamental data analysis tools, I'm
   surprised and   disappointed that they are apparently not
   within Mma's capabilities. Am I   incorrect in my


and got back a cordial but noncommittal response:

   Most of what I can say about this is obvious.  Adding new
   functions and tuning the efficiency of existing functions is a
   constant area of work.  Just like everyone else, we are limited
   by time and resources, and must set priorities for which
   functions to add and which functions to investigate for

   All of these functions in Mathematica are based on subroutines
   from LINPACK and EISPACK, and will inherit the capabilities
   and limitations of those functions.  We am aware of other
   subroutine libraries that may be more efficient, and will
   investigate making improvements as time permits.

   I will forward your suggestions to the appropriate people. 

   Since making any changes or improvements in these areas is a
   formidable undertaking, I am unable to predict when or if any
   changes will be made. 


Being a good consumer, I want instant gratification and, therefore, was  
hoping that some kind soul out there had run into similar needs and had  
developed the requisite Mathematica code. Otherwise, I fear that I shall have  
to dig into the numerical aspects myself. Since the approaches which jump to  
mind feature SVDs it appears that I would have to start with generating a  
more efficient SVD. :(

Mayhaps, if any of y'all are concerned about the numerical matrix algebra  
aspects of Mma, you could drop a note to WRI and explain the urgency of their  
working on my problems. ;)

Any help would be much appreciated,

Mark Kotanchek
Signal Processing Dept - 363 ASB
Applied Research Lab/Penn State
P.O. Box 30
State College, PA 16804

e-mail:	kotanchek at (NeXTmail)
TEL:	(814)863-0682
FAX:	(814)863-0753

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