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 \(\mathcal{M}\) is invariant under the linear transformation \(A\) , then \(E A E=A E\) for every projection \(E\) on \(\mathcal{M}\) . Conversely, if \(E A E=A E\) for some projection \(E\) on \(\mathcal{M}\) , then \(\mathcal{M}\) is invariant under \(A\) .

Proof. Suppose that \(\mathcal{M}\) is invariant under \(A\) and that \(\mathcal{V}=\mathcal{M} \oplus \mathcal{N}\) for some \(\mathcal{N}\) ; let \(E\) be the projection on \(\mathcal{M}\) along \(\mathcal{N}\) . For any \(z=x+y\) (with \(x\) in \(\mathcal{M}\) and \(y\) in \(\mathcal{N}\) ) we have \(A E z=A x\) and \(E A E z=E A x\) ; since the presence of \(x\) in \(\mathcal{M}\) guarantees the presence of \(A x\) in \(\mathcal{M}\) , it follows that \(E A x\) is also equal to \(A x\) , as desired.

Conversely, suppose that \(\mathcal{V}=\mathcal{M} \oplus \mathcal{N}\) , and that \(E A E=A E\) for the projection \(E\) on \(\mathcal{M}\) along \(\mathcal{N}\) . If \(x\) is in \(\mathcal{M}\) , then \(E x=x\) , so that \[E A x=E A E x=A E x=A x,\] and consequently \(A x\) is also in \(\mathcal{M}\) . ◻

Theorem 2. If \(\mathcal{M}\) and \(\mathcal{N}\) are subspaces with \(\mathcal{V}=\mathcal{M} \oplus \mathcal{N}\) , then a necessary and sufficient condition that the linear transformation \(A\) be reduced by the pair \((\mathcal{M}, \mathcal{N})\) is that \(E A=A E\) , where \(E\) is the projection on \(\mathcal{M}\) along \(\mathcal{N}\) .

Proof. First we assume that \(E A=A E\) , and we prove that \(A\) is reduced by \((\mathcal{M}, \mathcal{N})\) . If \(x\) is in \(\mathcal{M}\) , then \(A x=A E x=E A x\) , so that \(A x\) is also in \(\mathcal{M}\) ; if \(x\) is in \(\mathcal{N}\) , then \(E x=0\) and \(E A x=A E x=A 0=0\) , so that \(A x\) is also in \(\mathcal{N}\) .

Next we assume that \(A\) is reduced by \((\mathcal{M}, \mathcal{N})\) , and we prove that \(EA = AE\) . Since \(\mathcal{M}\) is invariant under \(A\) , Theorem 1 assures us that \[E A E=A E;\] since \(\mathcal{N}\) is also invariant under \(A\) , and since \(1-E\) is a projection on \(\mathcal{N}\) , we have, similarly, \[(1-E) A(1-E)=A(1-E).\] From the second equation, after carrying out the indicated multiplications and simplifying, we obtain \(E A E=E A\) ; this concludes the proof of the theorem. ◻

EXERCISES

Exercise 1. 

  1. Suppose that \(E\) is a projection on a vector space \(\mathcal{V}\) , and suppose that scalar multiplication is redefined so that the new product of a scalar \(\alpha\) and a vector \(x\) is the old product of \(\alpha\) and \(Ex\) . Show that vector addition (old) and scalar multiplication (new) satisfy all the axioms on a vector space except \(1 \cdot x=x\) .
  2. 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 \(1 \cdot x=x\) ?

Exercise 2. 

  1. Suppose that \(\mathcal{V}\) is a vector space, \(x_{0}\) is a vector in \(\mathcal{V}\) , and \(y_{0}\) is a linear functional on \(\mathcal{V}\) ; write \(A x=[x, y_{0}] x_{0}\) for every \(x\) in \(\mathcal{V}\) . Under what conditions on \(x_{0}\) and \(y_{0}\) is \(A\) a projection?
  2. If \(A\) is the projection on, say, \(\mathcal{M}\) along \(\mathcal{N}\) , characterize \(\mathcal{M}\) and \(\mathcal{N}\) in terms of \(x_{0}\) and \(y_{0}\) .

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

Exercise 4. If \(A\) is a linear transformation, if \(E\) is a projection, and if \(F=1-E\) , then \[A=E A E+E A F+F A E+F A F\] Use this result to prove the multiplication rule for partitioned (square) matrices (as in Section: Matrices of transformations , Ex. 19).

Exercise 5. 

  1. If \(E_{1}\) and \(E_{2}\) are projections on \(\mathcal{M}_{1}\) and \(\mathcal{M}_{2}\) along \(\mathcal{N}_{1}\) and \(\mathcal{N}_{2}\) respectively, and if \(E_{1}\) and \(E_{2}\) commute, then \(E_{1}+E_{2}-E_{1} E_{2}\) is a projection.
  2. If \(E_{1}+E_{2}-E_{1} E_{2}\) is the projection on \(\mathcal{M}\) along \(\mathcal{N}\) , describe \(\mathcal{M}\) and \(\mathcal{N}\) in terms of \(\mathcal{M}_{1}\) , \(\mathcal{M}_{2}\) , \(\mathcal{N}_{1}\) , and \(\mathcal{N}_{2}\) .

Exercise 6. 

  1. Find a linear transformation \(A\) such that \(A^{2}(1-A)=0\) but \(A\) is not idempotent.
  2. Find a linear transformation \(A\) such that \(A(1-A)^{2}=0\) but \(A\) is not idempotent.
  3. Prove that if \(A\) is a linear transformation such that \(A^{2}(1-A)=A(1-A)^{2}=0\) , then \(A\) is idempotent.

Exercise 7. 

  1. Prove that if \(E\) is a projection on a finite-dimensional vector space, then there exists a basis \(\mathcal{X}\) such that the matrix \((e_{i j})\) of \(E\) with respect to \(\mathcal{X}\) has the following special form: \(e_{i j}=0\) or \(1\) for all \(i\) and \(j\) , and \(e_{i j}=0\) if \(i \neq j\) .
  2. An involution is a linear transformation \(U\) such that \(U^{2}=1\) . Show that if \(1+1 \neq 0\) , then the equation \(U=2 E-1\) establishes a one-to-one correspondence between all projections \(E\) and all involutions \(U\) .
  3. What do (a) and (b) imply about the matrix of an involution on a finite-dimensional vector space?

Exercise 8. 

  1. In the space \(\mathbb{C}^{2}\) of all vectors \((\xi_{1}, \xi_{2})\) let \(\mathcal{M}^{+}\) , \(\mathcal{N}_{1}\) , and \(\mathcal{N}_{2}\) be the subspaces characterized by \(\xi_{1}=\xi_{2}\) , \(\xi_{1}=0\) , and \(\xi_{2}=0\) , respectively. If \(E_{1}\) and \(E_{2}\) are the projections on \(\mathcal{M}^{+}\) along \(\mathcal{N}_{1}\) and \(\mathcal{N}_{2}\) respectively, show that \(E_{1} E_{2}=E_{2}\) and \(E_{2} E_{1}\) \(=E_{1}\) .
  2. Let \(\mathcal{M}^{-}\) be the subspace characterized by \(\xi_{1}=-\xi_{2}\) . If \(E_{0}\) is the projection on \(\mathcal{N}_{2}\) along \(\mathcal{M}^{-}\) , then \(E_{2} E_{0}\) is a projection, but \(E_{0} E_{2}\) is not.

Exercise 9. Show that if \(E\) , \(F\) , and \(G\) are projections on a vector space over a field whose characteristic is not equal to \(2\) , and if \(E+F+G=1\) , then \[E F=F E=E G=G E=F G=G F=0.\] Does the proof work for four projections instead of three?