Rank one decomposition of a positive semi-definite matrix with inequality trace constraints
6 views (last 30 days)
Show older comments
Suppose there is a square matrix A and a positive semi-definite matrix
, such that
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/528394/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/528399/image.png)
Is there any ways I could do the rank one decomposition of matrix X, such that for
,
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/528404/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/528409/image.png)
and keep the inquality constraints
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/528414/image.png)
Or at least hold for the most significant (largest eigenvalue)
?
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/528419/image.png)
Many thanks!
Accepted Answer
Matt J
on 23 Feb 2021
Edited: Matt J
on 23 Feb 2021
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
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/528434/image.png)
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
2 Comments
Matt J
on 23 Feb 2021
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.
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/528714/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/528719/image.png)
More Answers (0)
See Also
Categories
Find more on Eigenvalues in Help Center and File Exchange
Community Treasure Hunt
Find the treasures in MATLAB Central and discover how the community can help you!
Start Hunting!