1. ## determine: definite matrix

I do have the following matrix:

Code:
0, 2, 1
2, 1, 3
1, 3, 7
To determine whether this is pos,- neg, -semi,...-definite, I tried it
with the eigenvalues, as following:

Code:
0-λ, 2, 1
2, 1-λ, 3
1, 3, 7-λ
then how should I proceed?

2. Hello,

The eigenvalues are the values that make the characteristic polynomial equal to 0.
And the characteristic polynomial (in lambda) is the determinant of your second matrix.

3. thank you for your quick response!

calculating the determinant equals:

Code:
(-λ)(1-λ)(7-λ) + 12 -(1-λ) - 4(7-λ) -9(-λ) = 0
and then

Code:
-λ3 + 8λ2 + 7λ -17 =0
but how do I solve a polynomial of the 3rd degree in my case?
However isn'i it possible by factorising?

4. Originally Posted by vanilla1
the 3rd degree in my case?
However isn'i it possible by factorising?
employ the factor theorem

5. @pickslides:

I tried it with several substitutes, but don't get the right number !

6. Originally Posted by vanilla1
@pickslides:

I tried it with several substitutes, but don't get the right number !
I see. This equation looks to only have one real solution.

You may need to use a numercial method, try newton's method with starting guess around $x=4$

7. sorry, but I've never used newton's method.
I will just ask the next time my teacher, thanks anyway!

8. Originally Posted by vanilla1
sorry, but I've never used newton's method.
I will just ask the next time my teacher, thanks anyway!
It's clear from a graph that there are three real solutions: plot y = -x^3 + 8x^2 + 7x -17 - Wolfram|Alpha

However, there is no simple way of getting the exact value of these solutions (and even if you did the form would be completely unwieldy). As suggested, a numerical approach is required. Or alternatively, use a CAS to get approximate solutions. You would NOT be expected to find them by hand.

9. But we just need to know how many of them are positive

10. First of all, a real symmetric matrix has only real eigenvalues. This fact is not hard to prove.

We want to know if all the eigenvalues are positive/negative (then the matrix is positive/negative definite).

If all are non-negative/positive but not positive/negative (matrix is positive/negative semi definite). Look at the product of all the eigenvalues. If it is negative you can eliminate a few possibilities. Can you tell which ones and how?

Also I would like to remind you that for small matrices, like the one given here, we can directly use the definition of definiteness. After that you have to do either trial and error to disprove definiteness or some simple high school algebra to prove definiteness!

11. I notice that the problem does NOT require you to actually find the eigenvalues, just to determine if the matrix is positive semidefinite, etc.

I assume you are looking at the eigenvalues to determine if they are all positive, etc. Again, it is not necessary to find exact values for them, approximate will do.

Or you could not look at eigenvalues at all but use the basic definitions.

Can you determine if $\begin{bmatrix}x & y & z\end{bmatrix}\begin{bmatrix}0 & 2 & 1 \\ 2 & 1 & 3 \\ 1 & 3 & 7\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}$ is necessarily positive, negative, etc.?