# A linear algebra question for semi-Euclidean norm

Let us consider Minkowski inner product on $$\mathbb R^{1+n}$$, defined by $$\langle v,w \rangle = -v_0w_0+\sum_{j=1}^n v_j w_j\quad \,\forall\, v,w \in \mathbb R^{1+n}.$$ We say that a vector $$v$$ is null if $$\langle v,v\rangle =0$$. Moreover, given a symmetric matrix $$A$$, we say that $$A$$ is null positive definite if $$\langle Av,v\rangle >0$$ for all non-zero null vectors $$v$$, and null semi-positive definite if $$\langle Av,v\rangle \geq 0$$ for all null vectors $$v$$.

Suppose $$A(t)$$ is a family of symmetric matrices smoothly depending on $$t \in [0,1]$$ and suppose that $$A(t)$$ is null positive definite for all $$t \in (0,t_0)$$. Suppose also that $$A(t_0)$$ is null semi-positive definite but it is not null positive definite. Does it follow that there exists a null vector $$v$$ such that $$A(t_0) v =0?$$

Consider the case $$n = 1$$.
Let $$A = \begin{pmatrix} -1 \\ & -2t\end{pmatrix}$$
If $$v = (1,1)$$, then $$Av = (-1,-2t) \neq 0$$. If $$w = (1,-1)$$, then $$Aw = (-1, 2t) \neq 0$$.
But $$\langle Av,v\rangle = \langle Aw,w\rangle = 1-2t$$, and so $$A$$ is null positive definition for $$t \in (0,\frac12)$$, null semi-positive definition at $$t = 1/2$$, but there are no non-trivial null vectors in the kernel of $$A$$ at $$t = 1/2$$.