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From: "Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid>
Newsgroups: comp.soft-sys.matlab
Subject: Re: generate a positive semi-definite matrices??
Date: Sun, 10 Feb 2008 21:55:03 +0000 (UTC)
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James Tursa <aclassyguywithaknotac@hotmail.com> wrote in message 
<3dkuq3ls0516g4inapn6hvll7sac7k44np@4ax.com>...
> On Sun, 10 Feb 2008 09:17:02 +0000 (UTC), "Roger Stafford"
> <ellieandrogerxyzzy@mindspring.com.invalid> wrote:
> >  Second observation: There seem to be two definitions of a semi-definite 
> >matrix floating around.  One, as for example in Wikipedia, insists that it 
both 
> >be Hermitian and have non-negative eigenvalues.  The other definition, 
as for 
> >example in MathWorld and which I have used here, requires only that its 
> >Hermitian part, (M+M')/2, have non-negative eigenvalues.
> >Roger Stafford
> 
> Where, exactly, are you getting these definitions from? Can you post
> some web links? The Hermitian restriction does not agree with my
> understanding at all. For a matrix M to be positive semi-definite, it
> simply has to satisfy this (which I am sure you already know but I
> repeat for others):
> 
>    x' * M * x >= 0   for all non-zero vectors x
> 
> where x' is the complex conjugate transpose operator.
> 
> And testing the sign of all of the eigenvalues of M is a necessary and
> sufficient test for this. I am not sure what the M + M' calculation in
> your post (from other links) has to do with this. For example, see
> this wiki link which agrees with my understanding:
> 
> http://en.wikipedia.org/wiki/Positive-definite_matrix
> 
> Here is an example of a random matrix I generated in MATLAB that is
> not Hermitian but is positive semi-definite:
> 
>    0.95012928514718   0.48598246870930   0.45646766516834
>    0.23113851357429   0.89129896614890   0.01850364324822
>    0.60684258354179   0.76209683302739   0.82140716429525
> 
> A Hermitian matrix  (i.e., where isequal(M,M') is true ) *will* be
> positive semi-definite (an easy thing to prove), but it is not a
> necessary condition for positive semi-definiteness. Perhaps this is
> where the confusion is?
> 
> James Tursa
----------
  I dug up a few web sites pertinent to this matter, James.  You'll note that the 
first one is the site you mentioned.  In its first paragraph, which I quote 
below, it appears to restrict positive definite matrices to those that are 
Hermitian.  The same is true of the MathWorld site for semi-definite matrices, 
though elsewhere MathWorld states it differently.  The matter is explained in 
greater depth in the third site below which discusses non-Hermitian matrices 
in one of its paragraphs.  As it states in its concluding remark in this 
paragraph, "There is no agreement in the literature on the proper definition 
of positive-definite for non-Hermitian matrices."  There are also some very 
similar remarks to this last in your referenced Wikipedia site.

  In particular you will note that if you were to require that your expression 
x'*M*x above be real and non-negative for all complex vectors x, then M 
must necessarily be Hermitian.  If you only require that the real part of x'*M*x 
be non-negative for all complex x, then that would allow M to be non-
Hermitian.

  Here are the web sites and quotations which I refer you to:
-----------------------------------------------------
http://en.wikipedia.org/wiki/Positive-definite_matrix

"In linear algebra, a positive-definite matrix is a Hermitian matrix which in 
many ways is analogous to a positive real number."
-----------------------------------------------------
http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html

"A positive semidefinite matrix is a Hermitian matrix all of whose eigenvalues 
are nonnegative."
-----------------------------------------------------
http://www.halfvalue.com/wiki.jsp?topic=Positive-definite_matrix

"Non-Hermitian matrices
A real matrix M may have the property that xTMx > 0 for all nonzero real 
vectors x without being symmetric. The matrix

 [1 1
  0 1]

provides an example. In general, we have xTMx > 0 for all real nonzero 
vectors x if and only if the symmetric part, (M + MT) / 2, is positive definite.

The situation for complex matrices may be different, depending on how one 
generalizes the inequality z*Mz > 0. If z*Mz is real for all complex vectors z, 
then the matrix M is necessarily Hermitian. So, if we require that z*Mz be real 
and positive, then M is automatically Hermitian. On the other hand, we have 
that Re(z*Mz) > 0 for all complex nonzero vectors z if and only if the 
Hermitian part, (M + M*) / 2, is positive definite.

In summary, the distinguishing feature between the real and complex case is 
that, a bounded positive operator on a complex Hilbert space is necessarily 
Hermitian, or self adjoint. The general claim can be argued using the 
polarization identity. That is no longer true in the real case.

There is no agreement in the literature on the proper definition of positive-
definite for non-Hermitian matrices."
-----------------------------------------------------

Roger Stafford