MATRICES AND VECTOR SPACES
also. Another, rather more general, expression that is also real is theHermitian
form
H(x)≡x†Ax, (8.112)whereAis Hermitian (i.e.A†=A) and the components ofxmay now be complex.
It is straightforward to show thatHis real, since
H∗=(HT)∗=x†A†x=x†Ax=H.With suitable generalisation, the properties of quadratic forms apply also to Her-
mitian forms, but to keep the presentation simple we will restrict our discussion
to quadratic forms.
A special case of a quadratic (Hermitian) form is one for whichQ=xTAxis greater than zero for all column matricesx. By choosing as the basis the
eigenvectors ofAwe haveQin the form
Q=λ 1 x^21 +λ 2 x^22 +λ 3 x^23.The requirement thatQ>0 for allxmeans that all the eigenvaluesλiofAmust
be positive. A symmetric (Hermitian) matrixAwith this property is calledpositive
definite.If,instead,Q≥0 for allxthen it is possible that some of the eigenvalues
are zero, andAis calledpositive semi-definite.
8.17.1 The stationary properties of the eigenvectorsConsider a quadratic form, such asQ(x)=〈x|Ax〉, equation (8.105), in a fixed
basis. As the vectorxis varied, through changes in its three componentsx 1 ,x 2
andx 3 , the value of the quantityQalso varies. Because of the homogeneous
form ofQwe may restrict any investigation of these variations to vectors of unit
length (since multiplying any vectorxby any scalarksimply multiplies the value
ofQby a factork^2 ).
Of particular interest are any vectorsxthat make the value of the quadraticform a maximum or minimum. A necessary, but not sufficient, condition for this
is thatQis stationary with respect to small variations ∆xinx, whilst〈x|x〉is
maintained at a constant value (unity).
In the chosen basis the quadratic form is given byQ=xTAxand, usingLagrange undetermined multipliers to incorporate the variational constraints, we
are led to seek solutions of
∆[xTAx−λ(xTx−1)] = 0. (8.113)This may be used directly, together with the fact that (∆xT)Ax=xTA∆x,sinceA
is symmetric, to obtain
Ax=λx (8.114)