Hermitian forms, Equivalence, Conjunctive Hermitian matrices, Reduction to canonical form, Definite and semi-definite forms

Hermitian Form. The form in n variables defined by

where matrix H is n-square and Hermitian and the components of vector X are complex numbers. The rank of H is called the rank of the form. If the rank r is less than n the form is called singular. If the rank is equal to n it is non-singular. The value h of a Hermitian form is always real.

If H and X are real, the Hermitian form reduces to the quadratic form as a special case.i.e. if H and X are real, (1) above is a real quadratic form. As a consequence, the theorems for Hermitian forms are analogous to those for quadratic forms.

Hermitian forms arise in different places, including atomic physics.

Linear transformations on Hermitian forms. The non-singular linear transformation X = BY carries the Hermitian form into another Hermitian form .

Equivalence of Hermitian forms. Two Hermitian forms are said to be equivalent if and only if there exists a non-singular linear transformation X = BY that carries one of the forms into the other.

Conjunctive Hermitian matrices. Two n-square Hermitian matrices A and B are called congunctive if there exists a non-singular matrix P such that .

Two Hermitian forms are equivalent if and only if their matrices are conjunctive.

The rank of an Hermitian form is invariant under a non-singular transformation of the variables.

Reduction to canonical form. A Hermitian form of rank r can be reduced to the canonical form

of index p and signature p - (r - p) .

Two Hermitian forms are equivalent if and only if they have the same rank and the same index or the same rank and the same signature.

Definite and semi-definite forms.

Positive definite form. A nonsingular Hermitian form h = in n variables is called positive definite if its rank and index are equal. Because it is nonsingular its rank r is equal to n. Thus, a positive definite Hermitian form can be reduced to . Its value h is greater than zero for any value of the X’s except the trivial case

Positive semi-definite form. A singular Hermitian form h = in n variables is called positive semi-definite if its rank and index are equal. Because it is singular its rank r is less than n. Thus, a positive semi-definite Hermitian form can be reduced to where r < n. Its value h is greater than or equal to zero for any value of the X’s.

Definite and semi-definite matrices. The matrix H of an Hermitian form

is called positive definite or positive semi-definite according as the Hermitian form is positive definite or positive semi-definite.

An Hermitian form is positive definite if and only if there exists a nonsingular matrix C such that

.

If an Hermitian matrix H is positive definite, every principal minor of H is positive, and conversely.

If an Hermitian matrix H is positive semi-definite, every principal minor of H is non-negative, and conversely.

References.

  Ayres. Matrices (Schaum).