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# Matrix Representation of the Approximation Problem

Module by: Mark A. Davenport. E-mail the author

Suppose our inner product space V=RMV=RM or CMCM with the standard inner product (which induces the 22-norm).

Re-examining what we have just derived, we can write our approximation x^=Px=Vcx^=Px=Vc, where VV is an M×NM×N matrix given by

V = v 1 v 2 v N V = v 1 v 2 v N
(1)

and cc is an N×1N×1 vector given by

c 1 c 2 c N . c 1 c 2 c N .
(2)

Given xRMxRM (or CMCM), our search for the closest approximation can be written as

min c x - V c 2 min c x - V c 2
(3)

or as

min c , e e 2 2 subjectto x = V c + e min c , e e 2 2 subjectto x = V c + e
(4)

Using VV, we can replace G=VHVG=VHV and b=VHxb=VHx. Thus, our solution can be written as

c = ( V H V ) - 1 V H x , c = ( V H V ) - 1 V H x ,
(5)

which yields the formula

x ^ = V ( V H V ) - 1 V H x . x ^ = V ( V H V ) - 1 V H x .
(6)

The matrix V=(VHV)-1VHV=(VHV)-1VH is known as the “pseudo-inverse.” Why the name “pseudo-inverse”? Observe that

V V = ( V H V ) - 1 V H V = I . V V = ( V H V ) - 1 V H V = I .
(7)

Note that x^=VVxx^=VVx. We can verify that VVVV is a projection matrix since

V V V V = V ( V H V ) - 1 V H V ( V H V ) - 1 V H = V ( V H V ) - 1 V H = V V V V V V = V ( V H V ) - 1 V H V ( V H V ) - 1 V H = V ( V H V ) - 1 V H = V V
(8)

Thus, given a set of NN linearly independent vectors in RMRM or CMCM (N<MN<M), we can use the pseudo-inverse to project any vector onto the subspace defined by those vectors. This can be useful any time we have a problem of the form:

x = V c + e x = V c + e
(9)

where xx denotes a set of known “observations”, VV is a set of known “expansion vectors”, cc are the unknown coefficients, and ee represents an unknown “noise” vector. In this case, the least-squares estimate is given by

c = V x , x ^ = V V x . c = V x , x ^ = V V x .
(10)

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