TITLE: Can a symmetric matrix always be represented as the sum of a positive-definite and negative-definite matrix?
QUESTION [8 upvotes]: I was wondering if it is possible to decompose any symmetric matrix into a positive definite and a negative definite component. I can't seem to think of a counterexample if the statement is false.

REPLY [2 votes]: This answer is years late but I like this proof. 
The set of positive definite matrices is open in the set of symmetric matrices and must thus contain $\frac{n(n+1)}{2}$ matrices which are linearly independent. Take that as a basis for the vector space of symmetric matrices. Then every element can be written as a linear combination of positive definite matrices, and we can simply group the basis elements by the positive coefficients and negative coefficients, giving us that a symmetric matrix M can be written as the sum of a positive definite matrix and the negation of a positive definite matrix.