Geometric Meaning
If we let the equation be with symmetric matrix A, then the geometric meaning is as follows.
If all eigenvalues of A are non-zero, then it is an ellipsoid or a hyperboloid. If all the eigenvalues are positive, then it is an ellipsoid; if all the eigenvalues are negative, it is an image ellipsoid; if some eigenvalues are positive and some are negative, then it is a hyperboloid.
If there exist one or more eigenvalues λi = 0, then if the corresponding bi ≠ 0, it is a paraboloid (either elliptic or hyperbolic); if the corresponding bi = 0, the dimension i degenerates and does not get into play, and the geometric meaning will be determined by other eigenvalues and other components of b. When it is a paraboloid, whether it is elliptic or hyperbolic is determined by whether all other non-zero eigenvalues are of the same sign: if they are, then it is elliptic; otherwise, it is hyperbolic.
Read more about this topic: Quadratic Form
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