I don't think your eigenvalue calculations are correct. Double-check those. Also, a matrix is positive semi-definite if and only if all the eigenvalues are what?
Given matrix:
| b 0 0 |
| 0 1 a |
| 0 a 1 |
I've calculated the eigenvalues as b, 0 and 1.
Now I'm being asked to find the values of a and b for which the matrix A is positive semi-definite.
I'm not quite sure how to go about it, any ideas? I just need a starting block
Well I got this far and from here I'd multiply that out, and I've done it over and over again but and it's just a complete mess. I can't seem to simplify and nothing cancels out. So I guess in short, no, I can't continue. Thanks for your help too, I really appreciate it.
So then we obtain eigenvalues of λ = b, and λ = 1 - a.
The next stage is obtaining the values for which matrix A is positive semi-definite.
And I got this far:
bx1^2 + x2^2 + x3^2 + 2ax2x3 = 0
Meaning the condition is:
bx1^2 + x2^2 + x3^2 + 2ax2x3 >= 0
And the question states that a>= -1 and b >= 0, so I'm assuming if all this is true then the condition is satisfied?
(I don't know how to do this coding either for the equations, sorry :P)
By that, I assume you mean taking those values, but them in the condition:
bx1^2 + x2^2 + x3^2 + 2ax2x3 >= 0
And seeing if it still holds true for positive semi-definite?
What is confusing me is the question not so much the maths. It's asking "for what values of a and b satisfy the positive semi-definite condition for matrix A". And so a sufficient answer to this question should surely then be:
b >= 0, a >= -1 and a <= 1 ???
This question has popped up in a different buit instead of starting a new thread I'll stick with this to make it easier.
Given matrix A, where a>= -1 and b >= 0
|b 0 0|
|0 1 a|
|0 a 1|
Can matrix A be negative semi-definite?
Well,
x'Ax <= 0 for this to be true
bx1^2 + x2^2 + x3^2 + 2ax2x3 <= 0
Assuming a takes a negative value, then this could be true. Is this correct?