Quadratic form diagonalization over Z5
Task number: 4477
Solution
The matrix of a given quadratic form (with respect to the standard basis) is \(\begin{pmatrix} 2 &4 &3\\ 4 &4 &3\\ 3 &3 &0 \end{pmatrix} \).
Now diagonalize it by using row and column transformations and on the right side perform only row transforms.
\(\left(\begin{array}{ccc|ccc} 2& 4& 3& 1& 0& 0\\ 4& 4& 3& 0& 1& 0\\ 3& 3& 0& 0& 0& 1 \end{array}\right)\sim \left(\begin{array}{ccc|ccc} 2& 0& 3& 1& 0& 0\\ 0& 1& 2& 3& 1& 0\\ 3& 2& 0& 0& 0& 1 \end{array}\right)\sim \left(\begin{array}{ccc|ccc} 2& 0& 0& 1& 0& 0\\ 0& 1& 2& 3& 1& 0\\ 0& 2& 3& 1& 0& 1 \end{array}\right)\sim \left(\begin{array}{ccc|ccc} 2& 0& 0& 1& 0& 0\\ 0& 1& 0& 3& 1& 0\\ 0& 0& 4& 0& 3& 1 \end{array}\right) \)
The rows of the right-hand side of the resulting matrix form the requested basis.
Answer
The basis is formed e.g. by vectors \((1, 0, 0)^{\mathrm T}, (3, 1, 0)^{\mathrm T}\) and \( (0, 3, 1)^{\mathrm T}\).
