Mathematical Foundations - Models for Bonding in Chemistry - page 8

0 1 0 1

1
2

1
2

1
2

1
2

!
¼

@ A @ A

10
01

þ ¼

P 1 P 2 þ ¼ 1 ð 1 : 39 Þ

1
2

1
2

1
2

1
2

Inmathematics, matrices having these properties (idempotency, mutual
exclusivity, completeness 3 ) are called projectors. In fact, acting on matrix
C of Equation (1.21)

P 1 C ¼ P 1 c 1 þ P 1 c 2 ¼ c 1 ð 1 : 40 Þ

since:

0 1 0 1 0 1

0
@

1
A

1
2

1
2

1 1

2

1 1

1
2

1
2

p

p þ p p

@ A @ A @ A

P 1 c 1 ¼ ¼ ¼ ¼ c 1 ð 1 : 41 Þ

1
2

1
2

1
2

1
2

1 1

2

1 1

p

p þ p p

0
@

1
A

0
@

1
A

0
@

1
A

1
2

1
2

1 1

2

1

1
2

1
2

p

p þ p

!
¼

0
0

P 1 c 2 ¼ ¼ ¼ 0

1
2

1
2

1 1

2

1 1

2

1

p p þ p

ð 1 : 42 Þ

so that, acting on the complete matrix C of the eigenvectors, P 1 selects its
eigenvector c 1 , at the same time annihilating c 2 . In the same way:

P 2 C ¼ P 2 c 1 þ P 2 c 2 ¼ c 2 ð 1 : 43 Þ

This makes evident the projector properties of matrices P 1 and P 2 .
Furthermore, matrices P 1 and P 2 allow one to write matrix A in the so-
called canonical form:

A ¼ l 1 P 1 þ l 2 P 2 ð 1 : 44 Þ

3 Often referred to as resolution of the identity.