QUADRATIC FORMS AND DEFINITE MATRICES

← Back to document page

QUADRATIC FORMS AND DEFINITE MATRICES1. DEFINITION AND CLASSIFICATION OF QUADRATIC of a QUADRATIC A denote an n x n symmetric matrix with real entries andlet x denote an n x 1 column vector. Then Q = x Ax is said to be aquadratic form . Note thatQ=x Ax=( ) a11 ann (x1xn)=(x1,x2, ,xn) anixi =a11x21+a12x1x2+...+a1nx1xn+a21x2x1+a22x 22+...+a2nx2xn+...+...+...+an1xnx1+an2xn x2+...+annx2n= i jaijxixj(1)For example, consider the matrixA=[1221]and the vector x. Q is given byQ=x Ax=[x1x2][1221][x1x2]=[x1+2x22x1+x2][x1x 2]=x21+2x1x2+2x1x2+x22=x21+4x1x2+ of the QUADRATIC form Q =x Ax:A QUADRATIC form is said to be:a:negative DEFINITE :Q<0whenx6=0b:negative semidefinite:Q 0for all x andQ=0for somex6=0c:positive DEFINITE :Q>0whenx6=0d:positive semidefinite:Q 0for all x and Q = 0 for somex6=0e:indefinite:Q>0for some x andQ<0for some otherxDate: September 14, FORMS AND DEFINITE MATRICESConsider as an example the 3x3 diagonal matrix D below and a general 3 element vector 100020004 The general QUADRATIC form is given byQ=x Ax=[x1x2x3] 100020004 x1x2x3 =[x12x24x3] x1x2x3 =x21+2x22+4x23Note that for any real vector x6=0, that

A negative semi-definite quadratic form is bounded above by the plane x = 0 but will touch the plane at more than the single point (0,0). It will touch the plane along a line. Figure 4 shows a negative-definite quadratic form. An indefinite quadratic form will notlie completely above or below the plane but will lie above

  Form, Quadratic, Matrices, Definite, Quadratic forms and definite matrices