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Simple formula for MoorePenrose pseudoinverse
Posted:
Dec 2, 2011 3:42 AM


Engineers and physicists find use for the socalled "MoorePenrose 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 MoorePenrose 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/MoorePenrose_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*FIn(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 pseudoinverse. 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



