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.
I do have the following matrix:
To determine whether this is pos,- neg, -semi,...-definite, I tried itCode:0, 2, 1 2, 1, 3 1, 3, 7
with the eigenvalues, as following:
then how should I proceed?Code:0-λ, 2, 1 2, 1-λ, 3 1, 3, 7-λ
thank you for your quick response!
calculating the determinant equals:
and thenCode:(-λ)(1-λ)(7-λ) + 12 -(1-λ) - 4(7-λ) -9(-λ) = 0
but how do I solve a polynomial of the 3rd degree in my case?Code:-λ3 + 8λ2 + 7λ -17 =0
However isn'i it possible by factorising?
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.
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!
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 is necessarily positive, negative, etc.?