# Data Matrices Definiteness In Semidefinite Programming

I'm little bit confused about the data matrices associated with an SDP program. Consider the following SDP program: I know that the Matrix $$X$$ must be positive semidefinite i.e. $$X \succeq 0$$ as the last constraint implies. My question is about the data matrices $$C$$ and $$A_i \forall i \in \{1, \cdots, m\}$$. Do they have to be positive semidefinite ? In other words, can we have a matrix $$C$$ for example that is either negative definite or indefinite ?

• Yes, positive semidefiniteness is only enforced on $X$, not $C$ Jul 19, 2021 at 4:20
• @user1936752 thank you for your answer. Do they have to be symmetrical (matrices $C$ and $A$)? Jul 19, 2021 at 4:53
• Due to the symmetry of $X$ one can always make $C$ and $A$ symmetric. Jul 19, 2021 at 8:06

We do not need $$C$$ to be positive semidefinite. It is an extension of linear programming.
$$C\cdot X=\sum_{i=1}^n \sum_{j=1}^n C_{ij}X_{ij}$$
Note that due to $$X$$ is symmetric we have $$C_{ij}X_{ij}+C_{ji}X_{ji}=(C_{ij}+C_{ji})X_{ij},$$
we can obtain the same expression using symmetric matrix by using $$\frac12(C+C^T)$$ instead. Hence, usually for simplicity, we just states that we want $$C$$ and $$A_i$$ to be symmetric.