Projections and invariance

We have already seen that the study of projections is equivalent to the study of direct sum decompositions. By means of projections we may also study the notions of invariance and reducibility.

Theorem 1. If a subspace is invariant under the linear transformation , then for every projection on . Conversely, if for some projection on , then is invariant under .

Proof. Suppose that is invariant under and that for some ; let be the projection on along . For any (with in and in ) we have and ; since the presence of in guarantees the presence of in , it follows that is also equal to , as desired.

Conversely, suppose that , and that for the projection on along . If is in , then , so that and consequently is also in . ◻

Theorem 2. If and are subspaces with , then a necessary and sufficient condition that the linear transformation be reduced by the pair is that , where is the projection on along .

Proof. First we assume that , and we prove that is reduced by . If is in , then , so that is also in ; if is in , then and , so that is also in .

Next we assume that is reduced by , and we prove that . Since is invariant under , Theorem 1 assures us that since is also invariant under , and since is a projection on , we have, similarly, From the second equation, after carrying out the indicated multiplications and simplifying, we obtain ; this concludes the proof of the theorem. ◻

EXERCISES

Exercise 1.

Suppose that is a projection on a vector space , and suppose that scalar multiplication is redefined so that the new product of a scalar and a vector is the old product of and . Show that vector addition (old) and scalar multiplication (new) satisfy all the axioms on a vector space except .
To what extent is it true that the method described in (a) is the only way to construct systems satisfying all the axioms on a vector space except ?

Exercise 2.

Suppose that is a vector space, is a vector in , and is a linear functional on ; write for every in . Under what conditions on and is a projection?
If is the projection on, say, along , characterize and in terms of and .

Exercise 3. If is left multiplication by on a space of linear transformations (cf. Section: Matrices of transformations , Ex. 5), under what conditions on is a projection?

Exercise 4. If is a linear transformation, if is a projection, and if , then Use this result to prove the multiplication rule for partitioned (square) matrices (as in Section: Matrices of transformations , Ex. 19).

Exercise 5.

If and are projections on and along and respectively, and if and commute, then is a projection.
If is the projection on along , describe and in terms of , , , and .

Exercise 6.

Find a linear transformation such that but is not idempotent.
Find a linear transformation such that but is not idempotent.
Prove that if is a linear transformation such that , then is idempotent.

Exercise 7.

Prove that if is a projection on a finite-dimensional vector space, then there exists a basis such that the matrix of with respect to has the following special form: or for all and , and if .
An involution is a linear transformation such that . Show that if , then the equation establishes a one-to-one correspondence between all projections and all involutions .
What do (a) and (b) imply about the matrix of an involution on a finite-dimensional vector space?

Exercise 8.

In the space of all vectors let , , and be the subspaces characterized by , , and , respectively. If and are the projections on along and respectively, show that and .
Let be the subspace characterized by . If is the projection on along , then is a projection, but is not.

Exercise 9. Show that if , , and are projections on a vector space over a field whose characteristic is not equal to , and if , then Does the proof work for four projections instead of three?

EXERCISES

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