Find the signature of the symmetric bilinear form given by $s(f,g) :=\int_{-1}^{1} f(x)g(x)(35x^4-30x^2+3)dx$

Question

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

Answer 1

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 $(2,1,1)$.