Dmak
Jul6-08, 08:41 PM
Hello I was reading through some research and I came across the proof of a lemma which I did not wholly understand. The problem statement is as follows:
Let F be a non-degenerate non-symmetic bilinear form in V. Then there exists a basis in V with respect to which F has one of the following matrices:
H_{\phi} =
\[ \left( \begin{array}{ccc}
cos \phi & sin \phi & 0 \\
-sin \phi & cos \phi & 0 \\
0 & 0 & 1 \end{array} \right)\], J = \[ \left( \begin{array}{ccc} 1& 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right)\], K = \[ \left( \begin{array}{ccc}
1 & 1 & 0 \\
-1 & 0 & 1 \\
0 & 1 & 0 \end{array} \right)\]
.
First we decompose F into the sum of a symmetric and a skew symmetric form, F = F_{+} + F_{-} . The proof then makes the assumption that dim \; ker \; F_{-}= 1. It seems that the dimension could equally be 2. Any idea's why this is the case? Or better yet can anyone offer another proof of this lemma? Thank you for your help.
Let F be a non-degenerate non-symmetic bilinear form in V. Then there exists a basis in V with respect to which F has one of the following matrices:
H_{\phi} =
\[ \left( \begin{array}{ccc}
cos \phi & sin \phi & 0 \\
-sin \phi & cos \phi & 0 \\
0 & 0 & 1 \end{array} \right)\], J = \[ \left( \begin{array}{ccc} 1& 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right)\], K = \[ \left( \begin{array}{ccc}
1 & 1 & 0 \\
-1 & 0 & 1 \\
0 & 1 & 0 \end{array} \right)\]
.
First we decompose F into the sum of a symmetric and a skew symmetric form, F = F_{+} + F_{-} . The proof then makes the assumption that dim \; ker \; F_{-}= 1. It seems that the dimension could equally be 2. Any idea's why this is the case? Or better yet can anyone offer another proof of this lemma? Thank you for your help.