Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  calculate the Clarke and Michel-Penot subdifferentials of the function which maps a symmetric matrix to its mth largest eigenvalue.  show these two subdifferentials coincide, and are identical for all choices of index m corresponding to equal eigenvalues. Our approach is via the generalized directional derivatives of the eigenvalue function, thereby completing earlier studies on the classical directional derivative.
['Jean-Baptiste Hiriart-Urruty', 'Adrian S. Lewis']
I am writing a research paper on a idea The Clarke and Michel-Penot Subdifferentials of the Eigenvalues of a Symmetric Matrix
[{"from": "human", "value": "I am writing a research paper on a idea The Clarke and Michel-Penot Subdifferentials of the Eigenvalues of a Symmetric Matrix"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  calculate the Clarke and Michel-Penot subdifferentials of the function which maps a symmetric matrix to its mth largest eigenvalue.  show these two subdifferentials coincide, and are identical for all choices of index m corresponding to equal eigenvalues. Our approach is via the generalized directional derivatives of the eigenvalue function, thereby completing earlier studies on the classical directional derivative."}]