Show LMI $F(x)\succ0$ is feasible if and only if the LMI $F(x) \succeq I_{n \times n}$ is feasible

Question

Let $F : V \to \Bbb S^{n\times n}$ be a linear map, where $V$ is a vector space and and $S^{n\times n}$ is the set of $n \times n$ symmetric matrices. Prove that the LMI $F(x) \succ 0$ is feasible if and only if the LMI $F(x) \succeq I_{n \times n}$ is feasible.

I kind know I show use eigenvalue of matrix $F(x)$ to find

$$\begin{bmatrix} F_1(x) & & \\ & F_2(x) & \\ & & F_3(x) \\ \end{bmatrix}$$

but got stuck here. Please Help!

Sorry, edited the question, it meant to be F – M.D. Mar 13 '18 at 17:56 — M.D., Mar 13 '18 at 17:56

score 1 · Answer 1 · answered Mar 14 '18 at 22:10

1

"If" should be easy. For "only if", note that the strict inequality $F(x)\succ0$ implies the existence of some $\varepsilon>0$ such that $F(x)\succeq\varepsilon I_{n\times n}$. Now, you only need to use linearity of $F$ to finish the proof. Can you take it from here?

answered Mar 14 '18 at 22:10

Nukular

340

not the strictness but continuous dependence on $x$ makes it possible to squeeze an epsilon there – percusse Mar 15 '18 at 16:06

Show LMI $F(x)\succ0$ is feasible if and only if the LMI $F(x) \succeq I_{n \times n}$ is feasible

1 Answers1