Let S be a semi-definite matrix, W_i an orthogonal matrix, i=1,2,3.
If S+W_1S^W_1^T+W_2S^W_2^T+W_3S^W_3^T=I is an identity, then give all the conditions that the eigenvalues of matrix S satisfy.
trace(S) = sum of eigenvalues of S =(1/4) *d, where 'd' is the dimension of the Identity matrix.
The above equality condition, together with non negativity constraints on individual eigenvalues, defines a Polyhedral set in R^d. A bounded Polyhedral set has no extreme directions and every point in the set can be expressed as a convex combination of a finite number of vertex (extreme) points.
You can have a look into the theory of Completely Positive Trace Preserving transformations, in this contexr. In particular, the works of Prof. M. D. Choi and properties of the Choi matrix.
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