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1. Determinant method.

For a given quadratic type.

Write its matrix and determine the positive definite nature of the quadratic form (or symmetric matrix) according to whether all the sequential principal and sub-formulas of the symmetric matrix are greater than zero.

2. Positive inertia index method.

For a given quadratic form, it is first reduced to a standard shape, and then the positive definite of the quadratic form is determined according to whether the number of square terms in the standard form with a positive coefficient is equal to n.

After the quadratic form is transformed into a standard form through the orthogonal transformation, the coefficient of the square term in the standard form is the eigenvalue of the quadratic matrix. Therefore, the eigenvalues of the quadratic matrix can be found first, and then the positive definite of the quadratic matrix can be determined according to whether the number of eigenvalues greater than zero is equal to n.

Definition: There is a real quadratic form, if there is f(x) 0 for any set of real numbers that are not all zeros, then this quadratic form is called a positive definite quadratic form, and its symmetry matrix a is called a positive definite matrix.

Methods for discriminating positive definite quadratic forms:

1): If the n coefficients in the quadratic standard form are greater than zero, then it is positively definite;

2): If the n eigenvalues of the symmetry matrix a of the quadratic type are greater than zero, then it is positively definite;

3): If the principal and sub-formulas of each order of the symmetric matrix a are all greater than zero, then it is positively definite.

Note: If A is an N-order square, then the 1st and 2nd orders are ,..located in the upper left corner of An-order sub-formula, i.e., the master and sub-formula of each order called a.

Positive definite discriminating of quadratic forms.

Solution: Method 1: Use the eigenvalues of the quadratic symmetry matrix to judge.

Start by writing the matrix of the quadratic type:

Since: all its eigenvalues can be obtained: 0, 0, 0

Therefore, the quadratic form is a positive definite quadratic type.

Method 2: Use the principal and sub-formulas of each order of the quadratic matrix to determine.

Since the matrix of this quadratic form is:

Because of its order of master and sub: 0, 0, 0

Therefore, the quadratic form is a positive definite quadratic type.