Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MATRICES AND VECTOR SPACES

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

a unitary matrix does not change the norm of a complex vector. The action of a

unitary matrix on a complex column matrix thus parallels that of an orthogonal

matrix acting on a real column matrix.

8.12.7 Normal matrices

A final important set of special matrices consists of thenormalmatrices, for which

AA†=A†A,

i.e. a normal matrix is one that commutes with its Hermitian conjugate.

We can easily show that Hermitian matrices and unitary matrices (or symmetric

matrices and orthogonal matrices in the real case) are examples of normal

matrices. For an Hermitian matrix,A=A†and so

AA†=AA=A†A.

Similarly, for a unitary matrix,A−^1 =A†and so

AA†=AA−^1 =A−^1 A=A†A.

Finally, we note that, ifAis normal then so too is its inverseA−^1 ,since

A−^1 (A−^1 )†=A−^1 (A†)−^1 =(A†A)−^1 =(AA†)−^1 =(A†)−^1 A−^1 =(A−^1 )†A−^1.

This broad class of matrices is important in the discussion of eigenvectors and

eigenvalues in the next section.

8.13 Eigenvectors and eigenvalues

Suppose that a linear operatorA transforms vectorsxin anN-dimensional

vector space into other vectorsAxin the same space. The possibility then arises

that there exist vectorsxeach of which is transformed byAinto a multiple of

itself. Such vectors would have to satisfy

Ax=λx. (8.67)

Any non-zero vectorxthat satisfies (8.67) for some value ofλis called an

eigenvectorof the linear operatorA,andλis called the correspondingeigenvalue.

As will be discussed below, in general the operatorA hasN independent

eigenvectorsxi, with eigenvaluesλi.Theλiare not necessarily all distinct.

If we choose a particular basis in the vector space, we can write (8.67) in terms

of the components ofAandxwith respect to this basis as the matrix equation

Ax=λx, (8.68)

whereAis anN×Nmatrix. The column matricesxthat satisfy (8.68) obviously