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
Famous quotes containing the words geometric and/or meaning:
“New York ... is a city of geometric heights, a petrified desert of grids and lattices, an inferno of greenish abstraction under a flat sky, a real Metropolis from which man is absent by his very accumulation.”
—Roland Barthes (1915–1980)
“Losing life is a trifle and I will have that courage when I need it. But to see the meaning of this life vanishing, our reason for existing disappearing, that is what I cannot stand. No one can live without reason.”
—Albert Camus (1913–1960)