Rank one decomposition of a positive semi-definite matrix with inequality trace constraints

Suppose there is a square matrix A and a positive semi-definite matrix , such that

Is there any ways I could do the rank one decomposition of matrix X, such that for ,

and keep the inquality constraints

Or at least hold for the most significant (largest eigenvalue) ?

Many thanks!

[KALYAN ACHARJYA]

What you have tried so far?

Answer 1

[Matt J]

Is there any ways I could do the rank one decomposition of matrix X, such that

The obvious answer seems to be to test each k to see which satisfies

and choose any subset of them.

Or at least hold for the most significant (largest eigenvalue) ?

I don't know why you think this is a special case if your first requirement. This is not possible in general, as can be seen from the example A=diag([1,-4]) and X=diag(4,1). In this case, you can only satisfy the requirement with the least significant eigenvalue,

x1 =

     2
     0


x2 =

     0
     1

>> x1.'*A*x1, x2.'*A*x2

ans =

     4


ans =

    -4

[Mingyang Sun]

Thank you for your relpying, is it possible that NO any satisfies inequality constraint? if so, is there a way to find one?

[Matt J]

If trace(A*X)<=0, There will always be some satisfying the constraint. Once you have the , you can check each one, as I mentioned.