Transcription of 1 Quadratic Forms - rmi.ge
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1 Quadratic Forms - rmi.ge
Reading[SB], Ch. , p. 375-3931 Quadratic FormsAquadratic functionf:R Rhas the formf(x) =a x2. Generalizationof this notion to two variables is thequadratic formQ(x1, x2) =a11x21+a12x1x2+a21x2x1+ each term has degree 2 (the sum of exponents is 2 for all summands).A Quadratic form of three variables looks asf(x1, x2, x3) =a11x21+a12x1x2+a13x1x3+a21x2x1+a22x22+a 23x2x3+a31x1x3+a32x3x2+ general Quadratic form ofnvariables is a real-valued functionQ:Rn Rof the formQ(x1, x2, .., xn) =a11x21+a12x1x2+..+a1nx1xn+a21x2x1+a22x2 2+..+a2nx2xn+..an1xnx1+an2xnx2+..+annx2n In shortQ(x1, x2, .., xn) = ni, we see a Quadratic form is determined by the matrixA= ann . Matrix Representation of Quadratic FormsLetQ(x1, x2, .., xn) = ni,jaijxixjbe a Quadratic form with matrixA. Easyto see thatQ(x1, .., xn) = (x1, .., xn) ann .1 EquivalentlyQ(x) =xT A Quadratic formQ(x1, x2, x3) = 5x21 10x1x2+x22whosesymmetric matrix isA=(5 5 5 1)is the product of three matrices(x1, x2, x3) (5 5 5 1) x1x2x3.
Reading [SB], Ch. 16.1-16.3, p. 375-393 1 Quadratic Forms A quadratic function f: R ! R has the form f(x) = a ¢ x2.Generalization of this notion to two variables is the quadratic form Q(x1;x2) = a11x 2 1 +a12x1x2 +a21x2x1 +a22x 2 2: Here each term has degree 2 (the sum of exponents is 2 for all summands).
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