quadratic forms on R-two

Ici, formulas to determiner positive-definiteness pour cette quadratique formes

v=\left(\!\!\begin{array}{c}x\\y\end{array}\!\!\right) , w=\left(\!\!\begin{array}{c}w^1\\w^2\end{array}\!\!\right)

  • uncoupled case:

\left(\begin{array}{cc}a&0\\ 0&b\end{array}\right) , \left(\begin{array}{cc}a&0\\ 0&a\end{array}\right)

[x,y]\left[\begin{array}{cc}a&0\\ 0&b\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]=ax^2+by^2\ge0

if and only if


  • semi-coupled case:

\left(\begin{array}{cc}a&1\\ 0&a\end{array}\right)

[x,y]\left[\begin{array}{cc}a&1\\ 0&a\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]=ax^2+xy+by^2=a(x+\frac{y}{2a})^2+(a-\frac{1}{4a^2})y^2\ge0

if and only if


  • rotato-(contractive/expansive) case:

\left(\begin{array}{cc}a&b\\ -b&a\end{array}\right)

[x,y]\left[\begin{array}{cc}a&b\\ -b&a\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]=a(x^2+y^2)\ge0

if and only if


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