Matrix representation of conic sections

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the matrix representation of conic sections permits the tools of <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a> to be used in the study of <a href="/facts/Conic_section/teqgLptX">conic sections</a>. It provides easy ways to calculate a conic section's <a href="/facts/Axis_of_rotation/PGcg3IY5">axis</a>, <a href="/facts/Vertex_(curve)/FkoCoHhV">vertices</a>, <a href="/facts/Tangent/sF0jH3EI">tangents</a> and the <a href="/facts/Pole_and_polar/cqg8T8qM">pole and polar</a> 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 <a href="/facts/Parallel_(geometry)/16r5yq7D">parallel</a> to the <a href="/facts/Coordinate_system/Mtd0q4SP">coordinate system</a>.
Conic sections (including <a href="/facts/Degenerate_conic/9Ek4ja1g">degenerate</a> ones) are the <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a> of points whose coordinates satisfy a second-degree <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> equation in two variables,

Q
        (
        x
        ,
        y
        )
        =
        A
        
          x
          
            2
          
        
        +
        B
        x
        y
        +
        C
        
          y
          
            2
          
        
        +
        D
        x
        +
        E
        y
        +
        F
        =
        0.
      
    
    {\displaystyle Q(x,y)=Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0.}

By an <a href="/facts/Abuse_of_notation/D4W6n85F">abuse of notation</a>, this conic section will also be called 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 when no confusion can arise.
This equation can be written in <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> notation, in terms of a <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric matrix</a> to simplify some subsequent formulae, as

 
 
 
 
 
 (
 
 
 
 x
 
 
 y
 
 
 
 )
 
 
 
 
 (
 
 
 
 A
 
 
 B
 
 /
 
 2
 
 
 
 
 B
 
 /
 
 2
 
 
 C
 
 
 
 )
 
 
 
 
 (
 
 
 
 x
 
 
 
 
 y
 
 
 
 )
 
 
 +
 
 
 (
 
 
 
 D
 
 
 E
 
 
 
 )
 
 
 
 
 (
 
 
 
 x
 
 
 
 
 y
 
 
 
 )
 
 
 +
 F
 =
 0.
 
 
 {\displaystyle {\begin{pmatrix}x&y\end{pmatrix}}{\begin{pmatrix}A&B/2\\B/2&C\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}+{\begin{pmatrix}D&E\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}+F=0.}

The sum of the first three terms of this equation, namely

A
        
          x
          
            2
          
        
        +
        B
        x
        y
        +
        C
        
          y
          
            2
          
        
        =
        
          
            (
            
              
                
                  x
                
                
                  y
                
              
            
            )
          
        
        
          
            (
            
              
                
                  A
                
                
                  B
                  
                    /
                  
                  2
                
              
              
                
                  B
                  
                    /
                  
                  2
                
                
                  C
                
              
            
            )
          
        
        
          
            (
            
              
                
                  x
                
              
              
                
                  y
                
              
            
            )
          
        
        ,
      
    
    {\displaystyle Ax^{2}+Bxy+Cy^{2}={\begin{pmatrix}x&y\end{pmatrix}}{\begin{pmatrix}A&B/2\\B/2&C\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}},}

is the <a href="/facts/Quadratic_form/l2n1jXPe">quadratic form</a> associated with the equation, and the matrix

A
          
            33
          
        
        =
        
          
            (
            
              
                
                  A
                
                
                  B
                  
                    /
                  
                  2
                
              
              
                
                  B
                  
                    /
                  
                  2
                
                
                  C
                
              
            
            )
          
        
      
    
    {\displaystyle A_{33}={\begin{pmatrix}A&B/2\\B/2&C\end{pmatrix}}}

is called the matrix of the quadratic form. The <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a> and <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of 
 
 
 
 
 A
 
 33
 
 
 
 
 {\displaystyle A_{33}}
 
 are both invariant with respect to rotation of axes and <a href="/facts/Translation_(geometry)/KlpWmnfX">translation</a> of the plane (movement of the origin).
The <a href="/facts/Quadratic_equation/9IMnkJAT">quadratic equation</a> can also be written as

 
 
 
 
 
 x
 
 
 
 T
 
 
 
 
 A
 
 Q
 
 
 
 x
 
 =
 0
 ,
 
 
 {\displaystyle \mathbf {x} ^{\mathsf {T}}A_{Q}\mathbf {x} =0,}

where 
 
 
 
 
 x
 
 
 
 {\displaystyle \mathbf {x} }
 
 is the <a href="/facts/Homogeneous_coordinates/TG9ebKN4">homogeneous coordinate vector</a> in three variables restricted so that the last variable is 1, i.e.,

 
 
 
 
 
 (
 
 
 
 x
 
 
 
 
 y
 
 
 
 
 1
 
 
 
 )
 
 
 
 
 {\displaystyle {\begin{pmatrix}x\\y\\1\end{pmatrix}}}

and where 
 
 
 
 
 A
 
 Q
 
 
 
 
 {\displaystyle A_{Q}}
 
 is the matrix

 
 
 
 
 A
 
 Q
 
 
 =
 
 
 (
 
 
 
 A
 
 
 B
 
 /
 
 2
 
 
 D
 
 /
 
 2
 
 
 
 
 B
 
 /
 
 2
 
 
 C
 
 
 E
 
 /
 
 2
 
 
 
 
 D
 
 /
 
 2
 
 
 E
 
 /
 
 2
 
 
 F
 
 
 
 )
 
 
 .
 
 
 {\displaystyle A_{Q}={\begin{pmatrix}A&B/2&D/2\\B/2&C&E/2\\D/2&E/2&F\end{pmatrix}}.}

The matrix 
 
 
 
 
 A
 
 Q
 
 
 
 
 {\displaystyle A_{Q}}
 
 is called the matrix of the quadratic equation. Like that of 
 
 
 
 
 A
 
 33
 
 
 
 
 {\displaystyle A_{33}}
 
, its determinant is invariant with respect to both rotation and translation.
The 2 × 2 upper left submatrix (a matrix of order 2) of 
 
 
 
 
 A
 
 Q
 
 
 
 
 {\displaystyle A_{Q}}
 
, obtained by removing the third (last) row and third (last) column from 
 
 
 
 
 A
 
 Q
 
 
 
 
 {\displaystyle A_{Q}}
 
 is the matrix of the quadratic form. The above notation 
 
 
 
 
 A
 
 33
 
 
 
 
 {\displaystyle A_{33}}
 
 is used in this article to emphasize this relationship.

Matrix representation of conic sections open-in-new

Matrix representation of conic sections