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5.7 Positive definite quadratic form

The standard form of the quadratic form is not unique, only that the number of terms contained in the standard form is determined (i.e. the rank of the quadratic form)

Theorem 9: Inertia theorem

Let’s say that the quadratic form f=xTAxf=x^TAxf=xTAx, whose rank is RRR, has two reversible transformations

$X = Cy, x = Pz$

make

$f=k_1y_1^2+k_2y_2^2+…. +k_ry_r^2(k_i\neq0)$

and

$f=\lambda_1z_1^2+\lambda_2z_2^2+… +\lambda_rz_r^2(\lambda_i\neq0)$

K1,… ,krk_1,… ,k_rk1,… , the number of positive numbers in Kr and λ1,…. , lambda r \ lambda_1,… , \ lambda_r lambda 1,… The number of positive numbers in λr is equal

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The number of positive coefficients in the standard form of the quadratic form is called the positive inertia coefficient of the quadratic form, and the number of negative coefficients is called the negative inertia coefficient

If the positive inertia coefficient of quadratic FFF is PPP and the rank is RRR, then the normal form of FFF can be determined as

$f=y_1^2+… +y_p^2-y_{p+1}^2-… -y_r^2$

Define 10

Let’s say the quadratic form f(x)=xTAxf(x)=x^TAxf(x)=xTAx

• If f(x)>0f(x)>0f(x)> 0f(x)>0 for any x≠0x\neq0x=0, FFF is said to be positive definite quadratic and the symmetric matrix A is said to be positive definite
• If f(x)<0f(x)<0f(x)< 0f(x)<0 for any x≠0x\neq0x=0, FFF is said to be of negative definite quadratic type and the symmetric matrix AAA is said to be negative definite

Theorem 10

The sufficient and necessary condition for the positive definite of the quadratic form f=xTAxf=x^TAxf=xTAx is that all the NNN coefficients of its standard form are positive, that is, all the NNN coefficients of its normal form are 1, that is, its positive inertia index is equal to NNN

inference

The necessary and sufficient condition for positive definite of symmetric matrix AAA is that all the eigenvalues of AAA are positive

Theorem 11: Hurwitz theorem

The necessary and sufficient conditions for the positive definite of symmetric matrix AAA are as follows: all of the hosts of A are positive, i.e

Symmetry is
$A$forNegative definiteThe sufficient and necessary condition of is: the odd order principal is negative, while the even order principal is positive, i.e

For example,

Cases of 17

Determine the positive characterization of the quadratic f=−5×2−6y2−4z2+ 4XY + 4xZf =-5x^2-6y^ 2-4Z ^2+ 4XY + 4xZF =−5×2−6y2− 4Z2 + 4XY +4xz

Answer:

The matrix AAA of quadratic FFF is

First order master

$|a_{11}|=-5<0$

Second order principal

Third order master form

It is found that the first and third orders are negative, and the second order is positive

According to theorem 11: Hurwitz theorem, get

FFF is negative definite quadratic

conclusion

Description:

• Refer to textbook “linear algebra” fifth edition tongji University mathematics department
• With the book concept explanation combined with some of their own understanding and thinking

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