 By Balakrishnan A.V.

ISBN-10: 0387905278

ISBN-13: 9780387905273

ISBN-10: 3540905278

ISBN-13: 9783540905271

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23). In fact, if y ERE, then EEty = y; and if y E R~ =NEt, then EEty = 0. This proves that EEt is the orthogonal projection onto RE. Also, if y E R~t = NE, then Et Ey = 0; and if y E REt' y = Et X for some x, then EtEy = EtEEtx = Etx- Et(I- EEt)x = Etx = y. 21) is satisfied. First, we note that (ii) implies that E* = (Et E)* E* = Et EE*; this shows that Nt = RE· ~REt· Now, if y ERE, then we can find x ENf such that y =Ex; thus Et y = Et Ex= x = (E)- 1 Ex= (E)- 1 y. Finally, if z E R~ = NE·' then by (i), EEt z = 0; using (ii), Etz = EtEEtz = 0.

5 23 Pseudo-inverses and the singular value decomposition It is well known from linear algebra that not all matrices have an inverse. Keeping in mind how useful inverses are, it is natural to search for some types of "generalized inverses" in case no inverse exists, which capture at least some of the nice properties. The right definition of a generalized inverse depends on the properties we are interested in, and we shall only define the so-called pseudo-inverse. Given an m x n matrix E, we consider it as a linear mapping of en into em .

To prove the continuity of the mapping y t-+ Tyf we first assume that f is continuous and has compact support, say, contained in the bounded interval [c, d]. For notational convenience we prove the continuity in Yo = 0. First, for y E]- ~' the function H ¢(x) = Tyf(x)- Ty 0 f(x) = f(x- y)- f(x) has support in the interval [- ~ + c, d + ~ ]. Vd-c+l. This proves the continuity in the considered special case. The case of an arbitrary function f E L 2 (JR) follows by an approximation argument, using 42 2.