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Math Help - Positive semi-definite values

  1. #1
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    Positive semi-definite values

    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
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  2. #2
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    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?
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  3. #3
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    If and only if the eigenvalues are non-negative. Also, it's given that b \geqslant 0.

    And the eigenvalues aren't correct? I've gone over them and get the same. Am I missing something here?
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  4. #4
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    Can you continue?
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  5. #5
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    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.
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  6. #6
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    You don't need to multiply everything out. You basically have two factors, and . Set them both equal to zero separately.
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  7. #7
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    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)
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  8. #8
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    So you get two solutions from that factor.
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  9. #9
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    So, just to clarify, λ = b, λ = 1-a and λ = 1+a ???

    Does the rest of what I said in my previous post make sense?
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  10. #10
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    Quote Originally Posted by Notreve View Post
    So, just to clarify, λ = b, λ = 1-a and λ = 1+a ???
    Correct.

    Does the rest of what I said in my previous post make sense?
    Perhaps, but I think you're doing more work than you need to do. Just set b >= 0, 1 - a >= 0, and 1 + a >= 0 (all of them must be true), and see what restrictions that places on a and b.
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  11. #11
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    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 ???
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  12. #12
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    Looks good to me!
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  13. #13
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    Hallelujah praise the lord! Thanks, you've been a great help. I got an exam in a months time and my university professor is away on business, which, as you can guess, is not ideal.

    Thanks again.
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  14. #14
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    You're very welcome. Hope you do well on your exam!
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  15. #15
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    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?
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