Chapter 14
Hermitian Operators and Unitary
Operators
Hermitian operators and unitary operators are quite often encountered in mathematical physics and, in particular, quantum physics. In this chapter we investigate their
basic properties. Both Hermitian operators and unitary operators fall under the
category of normal operators. The normal matrices are characterized by an important
fact that those matrices can be diagonalized by a unitary matrix. Moreover,
Hermitian matrices always possess real eigenvalues. This fact largely eases mathematical treatment of quantum mechanics. In relation to these topics, in this chapter
we investigate projection operators systematically. We find their important application to physicochemical problems in Part IV. We further investigate Hermitian
quadratic forms and real symmetric quadratic forms as an important branch of matrix
algebra. In connection with this topic, positive definiteness and non-negative property of a matrix are an important concept. This characteristic is readily applicable to
theory of differential operators, thus rendering this chapter closely related to basic
concepts of quantum physics.
14.1 Projection Operators
In Chap. 12 we considered the decomposition of a vector space to direct sum of
invariant subspaces. We also mentioned properties of idempotent operators. Moreover, we have shown how an orthonormal basis can be constructed from a set of
linearly independent vectors. In this section an orthonormal basis set is implied as
basis vectors in an inner product space V
n .
Let us start with a concept of an orthogonal complement. Let W be a subspace in
V
n . Let us think of a set of vectors jxi such that
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_14
547
Hermitian Operators and Unitary
Operators
Hermitian operators and unitary operators are quite often encountered in mathematical physics and, in particular, quantum physics. In this chapter we investigate their
basic properties. Both Hermitian operators and unitary operators fall under the
category of normal operators. The normal matrices are characterized by an important
fact that those matrices can be diagonalized by a unitary matrix. Moreover,
Hermitian matrices always possess real eigenvalues. This fact largely eases mathematical treatment of quantum mechanics. In relation to these topics, in this chapter
we investigate projection operators systematically. We find their important application to physicochemical problems in Part IV. We further investigate Hermitian
quadratic forms and real symmetric quadratic forms as an important branch of matrix
algebra. In connection with this topic, positive definiteness and non-negative property of a matrix are an important concept. This characteristic is readily applicable to
theory of differential operators, thus rendering this chapter closely related to basic
concepts of quantum physics.
14.1 Projection Operators
In Chap. 12 we considered the decomposition of a vector space to direct sum of
invariant subspaces. We also mentioned properties of idempotent operators. Moreover, we have shown how an orthonormal basis can be constructed from a set of
linearly independent vectors. In this section an orthonormal basis set is implied as
basis vectors in an inner product space V
n .
Let us start with a concept of an orthogonal complement. Let W be a subspace in
V
n . Let us think of a set of vectors jxi such that
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_14
547