Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.12 SPECIAL TYPES OF SQUARE MATRIX

Suppose thaty=Axis represented in some coordinate system by the matrix

equationy=Ax;then〈y|y〉is given in this coordinate system by

yTy=xTATAx=xTx.

Hence〈y|y〉=〈x|x〉, showing that the action of a linear operator represented by

an orthogonal matrix does not change the norm of a real vector.

8.12.5 Hermitian and anti-Hermitian matrices

AnHermitianmatrix is one that satisfiesA=A†,whereA†is the Hermitian conju-

gate discussed in section 8.7. Similarly ifA†=−A,thenAis calledanti-Hermitian.

A real (anti-)symmetric matrix is a special case of an (anti-)Hermitian matrix, in

which all the elements of the matrix are real. Also, ifAis an (anti-)Hermitian

matrix then so too is its inverseA−^1 ,since

(A−^1 )†=(A†)−^1 =±A−^1.

AnyN×NmatrixAcan be written as the sum of an Hermitian matrix and

an anti-Hermitian matrix, since

A=^12 (A+A†)+^12 (A−A†)=B+C,

where clearlyB=B†andC=−C†. The matrixBis called the Hermitian part of

A,andCis called the anti-Hermitian part.

8.12.6 Unitary matrices

AunitarymatrixAis defined as one for which

A†=A−^1. (8.66)

Clearly, ifAis real thenA†=AT, showing that a real orthogonal matrix is a

special case of a unitary matrix, one in which all the elements are real. We note

that the inverseA−^1 of a unitary is also unitary, since

(A−^1 )†=(A†)−^1 =(A−^1 )−^1.

Moreover, since for a unitary matrixA†A=I, we have

|A†A|=|A†||A|=|A|∗|A|=|I|=1.

Thus the determinant of a unitary matrix has unit modulus.

A unitary matrix represents, in a particular basis, a linear operator that leaves

the norms (lengths) of complex vectors unchanged. Ify=Axis represented in

some coordinate system by the matrix equationy=Axthen〈y|y〉is given in this

coordinate system by

y†y=x†A†Ax=x†x.