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

a,b\ge0

  • 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

a\ge\frac{1}{2}

  • 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

a>0

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s