Thread: 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

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?

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?

Can you continue?

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.

You don't need to multiply everything out. You basically have two factors, and . Set them both equal to zero separately.

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)

So you get two solutions from that factor.

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?

Originally Posted by Notreve

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.

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 ???

Looks good to me!

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.

You're very welcome. Hope you do well on your exam!

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?