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Concept
Matrix representation of conic sections
Summary
In mathematics, the matrix representation of conic sections permits the tools of linear algebra to be used in the study of conic sections. It provides easy ways to calculate a conic section's axis, vertices, tangents and the pole and polar relationship between points and lines of the plane determined by the conic. The technique does not require putting the equation of a conic section into a standard form, thus making it easier to investigate those conic sections whose axes are not parallel to the coordinate system.
Conic sections (including degenerate ones) are the sets of points whose coordinates satisfy a second-degree polynomial equation in two variables,
By an abuse of notation, this conic section will also be called Q when no confusion can arise.
This equation can be written in matrix notation, in terms of a symmetric matrix to simplify some subsequent formulae, as
The sum of the first three terms of this equation, namely
is the quadratic form associated with the equation, and the matrix
is called the matrix of the quadratic form. The trace and determinant of are both invariant with respect to rotation of axes and translation of the plane (movement of the origin).
The quadratic equation can also be written as
where is the homogeneous coordinate vector in three variables restricted so that the last variable is 1, i.e.,
and where is the matrix
The matrix is called the matrix of the quadratic equation. Like that of , its determinant is invariant with respect to both rotation and translation.
The 2 × 2 upper left submatrix (a matrix of order 2) of AQ, obtained by removing the third (last) row and third (last) column from AQ is the matrix of the quadratic form. The above notation A33 is used in this article to emphasize this relationship.
Proper (non-degenerate) and degenerate conic sections can be distinguished based on the determinant of AQ:
If , the conic is degenerate.
If so that Q is not degenerate, we can see what type of conic section it is by computing the minor, :
Q is a hyperbola if and only if ,
Q is a parabola if and only if , and
Q is an ellipse if and only if .
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