Motivation: We wish to make an second order approximation of a nonconvex function $f(x), x \in \mathbb{R}^n$ such that it is convex, however we do not have guarantee that $\nabla^2 f(x)$ is convex
Given a matrix $A \in \mathbb{R}^{n \times n}$ not positive semidefinite
Does there exist a method to break $A$ that "preserves" $A$ as much as possible to obtain positive semidefinite matrix $M$ s.t. $A = M + N$?
If there is no method to tackle a general matrix, how about a symmetric matrix $A \in S^n$ where $S^n$ is the set of symmetric n by n real entries matrices?
Best Answer
In general, there is a decomposition of square matrices into a symmetric part and anti-symmetric part: $$A=M+N$$ where $M$ is symmetric and $N$ is anti-symmetric. This is a very general result, and this decomposition is in fact a Hilbert space decomposition so that $M$ is the "closest" symmetric matrix to $A$. It is easy to derive the forms of $M,N$: $$M=\frac{1}{2}(A+A^T), \ \mathrm{and} \ N=\frac{1}{2}(A-A^T).$$
A symmetric matrix $M$ then can be orthogonally diagonalized $M=Q^TDQ$, where the diagonal matrix $D=$ diag$(\lambda_1,...,\lambda_n)$ has real diagonal entries. Thus, we can write $D=D_{pos}+D_{neg}$, where $D_{pos}$ contains the positive eigenvalues and $D_{neg}$ the negative. Then we have $$A=Q^TD_{pos}Q+B$$ where obviously $Q^TD_{pos}Q$ is positive semi-definite and $B$ is everything else.
I'm unsure how much this matrix "preserves" $A$, as you say, but it is definitely the "positive semi-definite part of $A$"