Symmetrization of Basis Vectors A. Projection Operators

A linear operator Ρ is said to be idempotent if

P2 = P. (3.1)

If Ρ is nonsingular, multiplication from the left or right with P^{_ 1} shows that Ρ is the identity operator. T o exclude this possibility, it will be assumed that

det Ρ = 0. (3.2) T h e trivial case where Ρ is the zero operator will be eliminated by

stipulating that Ρ Φ 0.

Let

M ; ^s

be the matrix representative of a linear operator defined on a two-dimensional vector space. Evidently A is nonzero and satisfies (3.1) and (3.2) for any value of a. If a = 0, A is in diagonal form, but A cannot be diagonalized when α Φ 0. Henceforth, it will be assumed that the matrix representatives of all nonzero operators satisfying (3.1) and (3.2) can be diagonalized. In fact, it will be assumed that Ρ is hermitian:

Ρ = P\ (3.3) A nonzero linear operator that is idempotent, singular, and hermitian

is called a projection operator, or more precisely, a perpendicular projection

3. S Y M M E T R I Z A T I O N OF BASIS VECTORS 359 operator (12). T h e significance of this terminology will be disclosed in the following discussion.

A projection operator Ρ has only two distinct eigenvalues: 1 or 0.

For if p is any eigenvalue of P, it must satisfy (3.1), so that p = 0 or p = 1. T h e degeneracy of the eigenvalue 1 is called the rank of Ρ and denoted p(P); the degeneracy of the eigenvalue 0 is called the nullity of Ρ and denoted v(P). T h e rank of Ρ is equal to the number of linearly independent rows or columns in any matrix representative of P. If Ρ is defined on an w-dimensional vector space,

P( P ) = t r P , v(P)=n-p(P). (3.4)

Let S be an /z-dimensional vector space, {e¹, e² , eⁿ} an orthonormal basis for S, and Ρ a projection operator on S. If ξ and Ρξ are nonzero, then Ρξ and (/ — Ρ)ξ are eigenvectors of Ρ corresponding, respectively, to the eigenvalues 1 and 0. These assertions are immediate consequences of (3.1):

Ρ(Ρξ) = Ρ*ξ = Ρξ, P(i - Ρ)ξ = (P- Ρ2)ξ = ο .

There are p(P) linearly independent eigenvectors with the eigenvalue p = 1. These eigenvectors may be obtained by

(1) Forming the sequence of η vectors Pe^x , Pe² , Peⁿ . (2) Deleting the zero vector wherever it occurs in the sequence.

(3) deleting any nonzero vector that is a linear combination of vectors preceding it in the sequence.

T h e eigenvectors thus obtained are p(P)-fold degenerate, so that any set of p(P) linear combinations of these vectors is also a set of eigen-vectors with p = 1. Similar remarks apply to the v(P) eigeneigen-vectors of Ρ corresponding to the eigenvalue p = 0.

Consider now the set Sx , consisting of all vectors in S satisfying Ρη = η. This set is a subspace of S> since it contains the zero vector and Ctf)x + ε2η2 , whenever it contains η1 and η2 . T h e eigenvectors of Ρ corresponding to the eigenvalue 1 constitute a basis for S¹ , so that dim S^± = p(P). Similarly, the set S² , consisting of all vectors in S satisfying Ρξ = 0, is a subspace of S. T h e eigenvectors of Ρ corre-sponding to the eigenvalue 0 constitute a basis for S2 , so that dim S2 = v(P). T h e only vector of S belonging to both S1 and S2 is the zero vector. For if ξ is any such vector, Ρξ = ξ = 0. Furthermore, an arbitrary vector of S can be expressed as the sum of a vector in S^±

360 8. T H E A N A L Y S I S O F S Y M M E T R I C A L S P I N S Y S T E M S

and a vector in *S2 . T h e proof follows from the fact that every vector in S may be written in the form

ξ = Ρξ + (1 -Ρ)ξ. (3.5) Since Ρ(Ρξ) = Ρξ, and P(l - Ρ)ξ = 0, Ρξ is in S1, and (7 - Ρ)ξ is in

S2 , as asserted. This decomposition of S is often described by saying that S is the direct sum of S1 and S2 .

The significance of the term "projection operator'' becomes evident upon observing that the effect of Ρ on any ξ in S is to project ξ onto the />(P)-dimensional subspace Sx . For, by (3.5), Ρξ = Ρ2ξ + P(l - Ρ)ξ = Ρξ + 0. Similarly, the effect of 1 — Ρ is to project ξ onto the e i n -dimensional subspace S2 . T h e significance of the term*'perpendicular projection operator'' is revealed by the fact that any vector in S1 is orthogonal to any vector in S2 . This result is a consequence of the hermitian character of P, and the fact that P(l — P) = 0:

(Ρξ, [1 - Ρ]ξ) = (ξ, P[J - Ρ]ξ) = 0. (3.6) T o illustrate these results, let S be a real three-dimensional space and

Πι)·

i l 1 ° ~^l\ 1

P = ±\ 0 2 0 , 1 - P = *

\ - l 0 1/

It is easily verified that Ρ is a projection operator of rank 2. Operating on the elements of the basis with Ρ yields

Pex = ^(e1 — e3) , Pé>2 = e2 , Pe3 = —\(e1 — es).

Pes may be discarded, since it is a scalar multiple of Pe1 . Therefore, the (normalized) eigenvectors of Ρ corresponding to p = 1 are e² and

— £ 3 ) · In this case, the subspace S1 is the plane defined by these eigenvectors.

The remaining eigenvector of Ρ may be obtained by applying (7 — P ) to e1 , e2 , and £3 :

(/ - P)e¹ = 1(β^λ + *8) , ( / - P K = 0, ( 7 - P >3 = 1 ( ^ + e³).

Hence \^/2(e¹ + e³) is a normalized eigenvector with p = 0, and Α 9² consists of the line defined by this vector. If ξ = ξ1β¹ + ξ²β² -f- f3e3 is

I U I

0 0 0 | .

1 0 1

3. S Y M M E T R I Z A T I O N OF BASIS VECTORS 361 an arbitrary vector in S^y the component of ξ in S^± (i.e., the projection of ξ on

5Ί)

T h e preceding theory can be put into a more symmetrical form upon noting that if Ρ is a projection operator of rank p(P)^y then 1 — Ρ is a projection operator of rank p(l — P) = tr(7 — P ) = v(P). Since Ρ is a projection on S¹ , and 1 — Ρ a projection on 5² , the notation can be simplified by writing P¹ for P, and P² for 1 — P. These projection operators satisfy the relations

P1* = P1, P^{2 2} = P²,

ΛΡ

= P

P, = o.

Any operators Ay P , such that AB = ΒΑ = 0, are said to be mutually orthogonal. Obviously, mutually orthogonal operators commute.

T h e theory of projection operators is easily generalized. A set of / (1 < / < n) projection operators P¹, P², P^l, satisfying

PⁱP^j = 8ⁱⁱPⁱ (f,; = 1,2,...,/), (3.7)

P

+P

+ -+P

= 1 (3.8)

leads to a direct sum decomposition of S into subspaces Sx, *S2, Sl. T h e subspace contains all vectors in S satisfying P^ = f. T h e eigenvectors of Pi corresponding to the eigenvalue 1 are a basis for Si, so that dim = tr Pi . Every vector in S can be expressed as the sum of a vector from each Si. Indeed (3.8) shows that, for any ξ in Sy

ξ = Ρ,ξ + Ρ2ξ + - + Ρ,ξ. (3.9)

Furthermore, the only vector common to each S{ is the zero vector. For if ξ is contained in each S^{, then ξ = Ρ^χξ = Ρ²ξ = ··· = P ^ . Combining these equations with (3.9), it follows that Ιξ — ξ (l φ 1), whence £ = 0. Finally, bases for S¹ , *S² , S^l may be combined to give a set of η linearly independent vectors that is a basis for the whole space S.