quadratic forms on R-three


v=\left(\!\!\begin{array}{c}x\\y\\z\end{array}\!\!\right) , w=\left(\!\!\begin{array}{c}w^1\\w^2\\w^3\end{array}\!\!\right)

  • \left(\begin{array}{ccc}a&0&0\\ 0&b&0\\ 0&0&c\end{array}\right) , \left(\begin{array}{ccc}a&0&0\\ 0&b&0\\ 0&0&b\end{array}\right) , \left(\begin{array}{ccc}a&0&0\\ 0&a&0\\ 0&0&a\end{array}\right) 

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

if and only if

a,b,c\ge0 

  • \left(\begin{array}{ccc}a&0&0\\ 0&b&1\\ 0&0&b\end{array}\right) , \left(\begin{array}{ccc}a&0&0\\ 0&a&1\\ 0&0&a\end{array}\right)

[x,y,z]\left[\begin{array}{ccc}a&0&0\\ 0&b&1\\ 0&0&b\end{array}\right]\left[\begin{array}{c}x\\y\\z\end{array}\right]

=ax^2+by^2+yz+cz^2=ax^2+b(y+\frac{z}{2b})^2+(b-\frac{1}{4b^2})z^2\ge0

if and only if

a\ge0 , b\ge\frac{1}{\sqrt[3]4}

  • \left(\begin{array}{ccc}a&1&0\\ 0&a&1\\ 0&0&a\end{array}\right)
  • [x,y,z]\left[\begin{array}{ccc}a&1&0\\ 0&a&1\\ 0&0&a\end{array}\right]\left[\begin{array}{c}x\\y\\z\end{array}\right]

    =ax^2+xy+ay^2+yz+az^2

    =a(x+\frac{y}{2a})^2+(a-\frac{1}{2a^2})y^2+a(z+\frac{y}{2a})^2\ge0

    if and only if

    a\ge\frac{1}{\sqrt[3]{2}}

  • \left(\begin{array}{ccc}a&0&0\\ 0&b&c\\ 0&-c&b\end{array}\right) , \left(\begin{array}{ccc}a&0&0\\ 0&a&b\\ 0&-b&a\end{array}\right) , \left(\begin{array}{ccc}a&0&0\\ 0&a&a\\ 0&-a&a\end{array}\right)
  • 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