Consider the quadratic form f(x,y,z)=beginbmatrixx & y & zendbmatrixAbeginbmatrixx y zendbmatrix where A is an invertible 3times3 symmetric matrix over Q. Assume that there exists (alpha,beta,gamma)inC^3setminus(0,0,0) such that f(alpha,beta,gamma)=0. Which of the following statements is necessarily true?

(D) If there exists (a,b,c)inQ^3setminus(0,0,0) such that f(a,b,c)=0, then A has a positive eigenvalue and a negative eigenvalue.

Q32Section B (Part B)32 of 120

Quadratic forms — what an isotropic vector forces on the signature

Consider the quadratic form

$f(x,y,z)=\begin{bmatrix}x & y & z\end{bmatrix}A\begin{bmatrix}x\ y\ z\end{bmatrix}$

where $A$ is an invertible $3\times3$ symmetric matrix over $\mathbb{Q}$ . Assume that there exists $(\alpha,\beta,\gamma)\in\mathbb{C}^3\setminus\{(0,0,0)\}$ such that $f(\alpha,\beta,\gamma)=0$ . Which of the following statements is necessarily true?

Pick an option to check your answer.

Keyboard: A–D answer · S solution · ←/→ previous/next