Articles of lmis

Strict Inequality in Homogenous LMI

I’m studying Stephen Boyd’s notes for EE 363, here. In particular, I’m working through lecture 15, slide 9 on strict linear matrix inequalities. An LMI is an expression of the form $G(x) = G_0 + x_1G_1 + \cdots + x_nG_n \geq 0$, where $G_i$ are symmetric $m \times m$ matrices and $x \in \mathbf{R}^n$. The […]

Maximize $\gamma$ such that $A+\gamma B\succeq 0$

Let $A \succ 0$ and $B$ be $n\times n$ symmetric matrices. Find the largest scalar $\gamma \geq 0$ such that $$A + \gamma B \succeq 0$$ Here is my (failed) attempt: The problem is: $$\max_{\gamma\ge 0} \gamma : v^\top(A+\gamma B)v \ge 0 \quad \forall v\in\mathbb R^n$$ Let $Au_i = \lambda u_i$ for $i=1,\dots,n$. And let […]

Maximize $\langle \mathrm A , \mathrm X \rangle$ subject to $\| \mathrm X \|_2 \leq 1$

Given $\mathrm A \in \mathbb R^{m \times n}$, $$\begin{array}{ll} \text{maximize} & \langle \mathrm A , \mathrm X \rangle\\ \text{subject to} & \| \mathrm X \|_2 \leq 1\end{array}$$ $\| \mathrm X \|_2 \leq 1$ is equivalent to $\sigma_{\max} (\mathrm X ) \leq 1$, which is equivalent to $\lambda_{\max} (\mathrm X^T \mathrm X) \leq 1$. Hence, $$1 […]

If $A^2\succ B^2$, then necessarily $A\succ B$

I remember reading somewhere about the following properties of non-negative definite matrix. But I don’t know how to prove it now. Let $A$ and $B$ be two non-negative definite matrices. If $A^2\succ B^2$, then it necessarily follows that $A\succ B$, but $A\succ B$ doesn’t necessarily leads to $A^2\succ B^2$. How can you prove it? Thanks!