Find the signature (n+,n-,no) of the following symmetric bilinear-form on the real vector space $V:=\mathbb{R}[X]_{\leq3}$ of Polynomials of degree ≤ 3
$s : V\times V\rightarrow \mathbb{R}$
$s(f,g) :=\int_{-1}^{1} f(x)g(x)(35x^4-30x^2+3)dx$
So far I found the matrix of the bilinear form given here:
\begin{bmatrix}0 & 0 & 0 &0\\0 & 0 & 0 &\frac{128}{315}\\0 & 0 & \frac{128}{315} & 0\\ 0 &\frac{128}{315} & 0 &\frac{128}{231}\end{bmatrix}
I know I could calculate the characteristic polynomial and then find the eigenvalues, however it seems really hard here. We had the spectral theorem, however i do not know how to actually use it on the example here, maybe you guys can help me solve this one. Kind regards
Best Answer
Instead of finding the eigenvalues, you can perform simultaneous row/column operations in order to bring the matrix into a diagonal form from which you can read the signature easily. This is similar to Gauss elimination, only whenever you do a row operation, you must also immediately do the corresponding column operation. In your case, we have
$$ \begin{bmatrix}0 & 0 & 0 &0\\0 & 0 & 0 &\frac{128}{315}\\0 & 0 & \frac{128}{315} & 0\\ 0 &\frac{128}{315} & 0 &\frac{128}{231}\end{bmatrix} \xrightarrow[C_4 \leftrightarrow C_2]{R_4 \leftrightarrow R_2} \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & \frac{128}{231} & 0 & \frac{128}{315} \\ 0 & 0 & \frac{128}{315} & 0 \\ 0 & \frac{128}{315} & 0 & 0 \end{bmatrix} \xrightarrow[C_4 = C_4 - C_2]{R_4 = R_4 - R_2} \\ \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & \frac{128}{231} & 0 & 0 \\ 0 & 0 & \frac{128}{315} & 0 \\ 0 & 0 & 0 & -\frac{128}{315} \end{bmatrix}. $$
Hence, the signature is $(2,1,1)$.