惯性定理的几何意义

Geometric Significance of Inertia Theorem

从几何角度揭示了惯性定理的结论,在二元和三元二次型中,化为标准形的过程,实际上可以看成通过建立新的坐标系,将二元或三元二次型所对应的等值线或者是等值面方程化为标准形的过程,由于等值线或等值面的图形是确定的,所以无论怎样建立坐标系,等值线或等值面的大致形状是不变的,进而得出二次型的正负惯性指数不变.

In this paper,the conclusion of the inertia theorem from the geometric point of view is revealed.In the binary and ternary quadratic form,the process of transforming them into the standard form can be seen as the process of transforming the isoline or isosurface equation corresponding to the binary or ternary quadratic form into the standard form by establishing a new coordinate system.Since the figure of the isoline or isosurface is determined,no matter how to establish it,the approximate shape of coordinate system,isoline or isosurface is invariable,and then the positive and negative inertia index of quadratic form is invariable。