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Let 
$A$ and 
$B$ be 
$n\,\times \,n$ complex Hermitian (or real symmetric) matrices with eigenvalues 
${{a}_{1}}\,\ge \,\cdots \,\ge \,{{a}_{n}}$ and 
${{b}_{1}}\,\ge \,\cdots \,\ge \,{{b}_{n}}$. All possible inertia values, ranks, and multiple eigenvalues of 
$A\,+\,B$ are determined. Extension of the results to the sum of 
$k$ matrices with 
$k\,>\,2$ and connections of the results to other subjects such as algebraic combinatorics are also discussed.
