Combinations of perpendicular projections

The sum theorem for perpendicular projections is now easy.

Theorem 1. If are (perpendicular) projections, then a necessary and sufficient condition that be a (perpendicular) projection is that whenever (that is, that the be pairwise orthogonal).

Proof. The proof of the sufficiency of the condition is trivial; we prove explicitly its necessity only, so that we now assume that is a perpendicular projection. If belongs to the range of some , then so that we must have equality all along. Since, in particular, we must have it follows that whenever . In other words, every in the range of is in the null-space (and, consequently, is orthogonal to the range) of every with ; using Section: Perpendicular projections , Theorem 3, we draw the desired conclusion. ◻

We end our discussion of projections with a brief study of order relations. It is tempting to write , for two perpendicular projections and , whenever . Earlier, however, we interpreted the sign , when used in an expression involving linear transformations and (as in ), to mean that is a positive transformation. There are also other possible reasons for considering to be smaller than ; we might have for all , or (see Section: Combinations of projections , (ii)). The situation is straightened out by the following theorem, which plays here a role similar to that of Section: Perpendicular projections , Theorem 3, that is, it establishes the coincidence of several seemingly different concepts concerning projections, some of which are defined algebraically while others refer to the underlying geometrical objects.

Theorem 2. For perpendicular projections and the following conditions are mutually equivalent.

Proof. We shall prove the implication relations

(i) (ii) (iii) (iv, a) (iv, b) (i).

(i) (ii). If , then, for all , (since and are perpendicular projections).

(ii) (iii). We assume that for all . Let us now take any in ; then we have so that , or , whence and consequently . In other words, in implies that is in , as was to be proved.

(iii) (iv, a). If , then is in for all , so that, for all , as was to be proved.

That (iv, a) implies (iv, b), and is in fact equivalent to it, follows by taking adjoints.

(iv) (i). If , then, for all , Since and are commutative projections, so also are and , and consequently is a projection. Hence This completes the proof of Theorem 2. ◻

In terms of the concepts introduced by now, it is possible to give a quite intuitive sounding formulation of the theorem of Section: Combinations of projections (in so far as it applies to perpendicular projections), as follows. For two perpendicular projections and , their sum, product, or difference is also a perpendicular projection if and only if is respectively orthogonal to, commutative with, or greater than .

EXERCISES

Exercise 1.

Give an example of a projection that is not a perpendicular projection.
Give an example of two projections and (they cannot both be perpendicular) such that and .

Exercise 2. Find the (perpendicular) projection of on the (one-dimensional) subspace of spanned by . (In other words: find the image of the given vector under the projection onto the given subspace.)

Exercise 3. Find the matrices of all perpendicular projections on .

Exercise 4. If , then a necessary and sufficient condition that be an involutory isometry is that be a perpendicular projection.

Exercise 5. A linear transformation is called a partial isometry if there exists a subspace such that whenever is in and whenever is in .

The adjoint of a partial isometry is a partial isometry.
If is a partial isometry and if is a subspace such that or according as is in or in , then is the perpendicular projection on .
Each of the following four conditions is necessary and sufficient that a linear transformation be a partial isometry.
1. ,
2. is a projection,
3. ,
4. is a projection.
If is a proper value of a partial isometry, then .
Give an example of a partial isometry that has as a proper value.

Exercise 6. Suppose that is a linear transformation on, and is a subspace of, a finite-dimensional vector space . Prove that if , then there exist linear transformations and on such that for all in . (Hint: let be a partial isometry such that or according as is in or in and such that .)

EXERCISES

Quick Links

Social Media