Transcription of QUADRATIC FORMS AND DEFINITE MATRICES
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QUADRATIC FORMS AND DEFINITE MATRICES
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+a22x2 2+..+a2nx2xn+..+..+..+an1xnx1+an2xnx2+.. +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 MATRICESC onsider as an example the 3x3 diagonal matrix D below and a
Definitions of definite and semi-definite matrices. Let A be a square matrix of order n and let x be an n elementvector. Then A is said to be positive semidefiniteiff for all vectors x. QUADRATIC FORMS AND DEFINITE MATRICES 5 FIGURE 3. Positive Semi-Definite Quadratic Form 2x2 1+4x x2 +2x22-5 0 5 x1-5
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