Perpendicular projections

We are now in a position to fulfill our earlier promise to investigate the projections associated with the particular direct sum decompositions . We shall call such a projection a perpendicular projection . Since is uniquely determined by the subspace , we need not specify both the direct summands associated with a projection if we already know that it is perpendicular. We shall call the (perpendicular) projection on along simply the projection on and we shall write .

Theorem 1. A linear transformation is a perpendicular projection if and only if . Perpendicular projections are positive linear transformations and have the property that for all .

Proof. If is a perpendicular projection, then Section: Adjoints of projections , Theorem 1 and the theorem of Section: Dual of a direct sum show (after, of course, the usual replacements, such as for and for ) that . Conversely if , then the idempotence of assures us that is the projection on along , where, of course, and are the range and the null-space of , respectively. Hence we need only show that and are orthogonal. For this purpose let be any element of and any element of ; the desired result follows from the relation The positive character of an satisfying follows from Applying this result to the perpendicular projection , we see that this concludes the proof of the theorem. ◻

For some of the generalizations of our theory it is useful to know that idempotence together with the last property mentioned in Theorem 1 is also characteristic of perpendicular projections.

Theorem 2. If a linear transformation is such that and for all , then .

Proof. We are to show that the range and the null-space of are orthogonal. If is in , then is in , since Hence with , so that and therefore . Consequently , so that is in ; this proves that . Conversely, if is in , so that , we write with in and in . Then (The reason for the last equality is that is in and therefore in .) Hence is in , so that , and therefore . ◻

We shall need also the fact that the theorem of Section: Combinations of projections remains true if the word "projection" is qualified throughout by "perpendicular." This is an immediate consequence of the preceding characterization of perpendicular projections and of the fact that sums and differences of self-adjoint transformations are self-adjoint, whereas the product of two self-adjoint transformations is self-adjoint if and only if they commute. By our present geometric methods it is also quite easy to generalize the part of the theorem dealing with sums from two summands to any finite number. The generalization is most conveniently stated in terms of the concept of orthogonality for projections; we shall say that two (perpendicular) projections and are orthogonal if . (Consideration of adjoints shows that this is equivalent to .) The following theorem shows that the geometric language is justified.

Theorem 3. Two perpendicular projections and are orthogonal if and only if the subspaces and (that is, the ranges of and ) are orthogonal.

Proof. If , and if and are in the ranges of and respectively, then If, conversely, and are orthogonal (so that ), then the fact that for in implies that for all (since is in and consequently in ). ◻

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