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Simple formula for Moore-Penrose pseudoinverse
Posted:
Dec 2, 2011 3:42 AM
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Engineers and physicists find use for the so-called
"Moore-Penrose pseudoinverse" of an operator F: H --> K
between Hilbert spaces H and K (which can be real or complex).
Typically, H and K are finite dimensional
(which will be assumed below for simplicity),
and F is viewed as a matrix.
A common notation for the Moore-Penrose pseudoinverse
(henceforth simply called "pseudoinverse" is F with a superscript + , F^+,
but for ASCII simplicity I will call the pseudoinverse G : K --> H.
Let Null(F) denote the nullspace of F and In(F) its initial space
(defined as the orthogonal complement of Null(F)). As usual, Range(F)
will denote its range. Note for future reference that In(F) = In(F*F),
where F* denotes the adjoint of F.
Most mathematicians would probably define the pseudoinverse G of F
as the unique operator K --> H such that:
(i) G is a left inverse for the restriction of F to its initial
space, i.e.,
GFx = x for all x in In(F),
which defines G on Range(F), and
(ii) G annihilates the orthogonal complement of Range(F).
Definitions in the engineering literature are typically expressed in terms
of matrices and are much more complicated, e.g.,
http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse.
It is almost immediate that the pseudoinverse G is given by
the simple formula
(*) G = (F*F)^(-1) F*,
where (F*F)^(-1) denotes the inverse of the restriction of F*F to
In(F*F) = In(F). To see this, just multiply on the right by F to get (i),
and check (ii) separately.
Oddly, formulas for G in the engineering literatures are typically much
more
complicated and generally require diagonalizing F*F, which can be difficult
to do explicitly for F*F large (because its characteristic polynomial
will typically have no algebraically expressible roots).
However, F*F^(-1) can be obtained without diagonalizing F*F. (Just
choose a basis for In(F), represent F*F|In(F) as a matrix with respect
to this basis, and invert.)
I was recently surprised to discover the simple formula (*) when
I had occasion to actually calculate a pseudo-inverse. It is inconceivable
to me that it could be previously unknown, and I would be grateful
for a reference, which I shall need for a physics paper.
For an audience of mathematicians, one could simply state (*)
with minimal comment, but for people who may be more comfortable thinking
in terms of of matrices, (*) may not seem obvious, and a reference is
probably
necessary. Of course, I could devote a few paragraphs to proving (*),
but the paper will have a space limitation which I'd rather use for
the subjects of the paper.
A reference accessible online would be the most helpful. I have
retired to a rural area of Nevada which is 200 miles from the nearest
research library, so to access a book I generally have to buy it.
I thank in advance anyone kind enough to be of help.
Stephen Parrott
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