Find a 2x2 matrix with positive eigenvalues, but a negative quadratic form for some x in $R^{2}$

Question

Find a 2x2 matrix with real and positive eigenvalues, but a negative definite quadratic form.

Also, find a 2x2 matrix with real and positive eigenvalues, but an indefinite quadratic form.

Isn't this not possible? If all eigenvalues are positive, isn't the matrix positive definite?

Answer 1

1)Real and positive eigenvalues, negative definite quadratic form.

You're looking for such a matrix $A=\begin{bmatrix}a && b \\ c && d\end{bmatrix}$, which satisfies some conditions:

$\textbf{a}$) $a<0$ and $\det A=ad-cb>0$ (negative definite quadratic form)

$\textbf{b)}$characteristic polynomial is $\chi(x)=(x-y)^2=x^2-2yx+y^2$ for some $y>0$ (positive eigenvalues).But you know that $\chi(x)=x^2-(a+d)x+ad-bc$.

Now you are looking for $a,b,c,d$. There are a lot of solutions, for example $a=-2$, $b=1$, $c=9$, $d=4$.

2)Real and positive eigenvalues, indefinite quadratic form.

If you put $a=0$ sometimes you can get indefinite quadratic form ($\textbf{b}$ the same like above). You can check that for example $b=-1$, $c=4$, $d=4$ is a solution (quadratic form if positive for $[x,y]=[1,1]$ and negative for $[x,y]=[-2,1]$.