Let $A$ be a $n\times n$ real symmetric non-singular matrix. Suppose there exists $x\in \mathbb{R^n}$ such that $x'Ax<0$. Then we can conclude that

  1. $\det(A)<0$
  2. $B=-A$ is positive definite.
  3. $\exists y\in \mathbb{R^n}: y'A^{-1}y<0$
  4. $\forall y\in \mathbb{R^n}: y'A^{-1}y<0.$

My work:

I don't know how to do it for $n$, so I cooked up a $2\times 2$ matrix $\begin{bmatrix}-1 &2\\ 2& -1 \end{bmatrix}$, with $x=(1,0)^t$ we get $x'Ax<0$. Now $\det(A)=-3<0$ and also we can find a $y$ such that $y'A^{-1}y<0.$ But the problem is, according to the question only one option is true. And I don't know if this problem can be solved using a particular $n=2$.

So how can I solve this? Any help would be great. Thanks.

  • $\begingroup$ $1$ is not true, for example $-I$ wouldn't work. $2$ isn't true either, since a matrix could have mixed eigenvalue signs. Finally, the fourth one is the same as negative definiteness of $A^{-1}$, which again is a wrong statement. Hence it looks like the third one is correct. $\endgroup$ – астон вілла олоф мэллбэрг Oct 31 '16 at 12:35
  • $\begingroup$ What do you know about symmetric real matrices? $\endgroup$ – principal-ideal-domain Oct 31 '16 at 12:36
  • $\begingroup$ nothing much. Just definitions and some preliminary facts. $\endgroup$ – Kushal Bhuyan Oct 31 '16 at 13:11

Take $n=2$,

For Option-$1$: Consider $A=\begin{bmatrix}-1 & 0\\ 0& -1 \end{bmatrix}$ then clearly $A$ is symmetric and non-singular matrix.

Moreover, there exists $x=(1,0)$ such that $x^TAx=-1$ but $det A=1$.

Option-$4$: Consider same matrix then $A^{-1}=\begin{bmatrix}-1 & 0\\ 0& -1 \end{bmatrix}$ but for $y=(0,0), y^TAy=0$

Option-$2$: Consider $A=\begin{bmatrix}-1 & 0\\ 0& 1 \end{bmatrix}$ then there exists $x=(-1,0)$ such that $x^TAx<0$ but $B=-A=\begin{bmatrix}1 & 0\\ 0& -1 \end{bmatrix}$ is not positive definite as determinant of $B$ is $-1$.

Hence, Option-$3$ is true.

  • $\begingroup$ Elimination of other options is not good Idea to say remaining option is correct. Can you please give me hint that how can we prove third option is correct? $\endgroup$ – ramanujan Nov 30 '18 at 0:02

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