Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$, how to call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$?

Question

Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$

How do people call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$?

Sum of positive components of $X$?

The positive semi definite part of $X$?

Answer 1

This is the positive part of the matrix, commonly denoted $X_+$. More generally, for any bounded self-adjoint operator $X$ on a Hilbert space, there are unique $X_-,X_+\geq 0$ such that $X=X_+-X_-$ and $X_+X_-=X_-X_+$. See also the answer to this question