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?

  • $\begingroup$ If a normal matrix has all positive eigenvalues, then it is positive definite, so you should look for a matrix that isn't normal, e.g., a matrix that isn't diagonalizable over $\mathbb{C}$. Up to similarity, what does a non-diagonalizable $2 \times 2$ matrix look like? $\endgroup$ – Branimir Ćaćić Jul 8 '14 at 23:28
  • $\begingroup$ So a non-symmetric one? For a negative definite, I managed to find [{1,-3},{0,1}]. But I'm stuck as to what technique I would use to construct an indefinite one. det (ad-bc) has to be negative. Whenever I select values to satisfty that, one of my eigenvalues turns out negative. $\endgroup$ – Bob Jul 8 '14 at 23:37
  • $\begingroup$ @Bob Definiteness implies semidefiniteness, so you're done. $\endgroup$ – Git Gud Jul 8 '14 at 23:51
  • $\begingroup$ I'm not sure what you mean.. $\endgroup$ – Bob Jul 8 '14 at 23:52
  • $\begingroup$ Whoops, nevermind. I see that I can use the same matrix for both questions. If x=[1,1] Q(x)<0 , but if x=[-1,1], Q(x)>0. $\endgroup$ – Bob Jul 9 '14 at 0:00

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 $[x,y]=[-2,1]$.


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