Definition 1. Consider the partitioned matrix
$$A=\left[\begin{array}{ll}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]$$

When $A_{11}$ is nonsingular, $A_{22}?A_{21}{A}_{11}^{?1}{A}_{12}$ is called the Schur complement of $A_{11}$ in $A$, denoted by $S_{ch}\left(A_{11}\right)$.

When $A_{22}$ is nonsingular, $A_{11}?A_{12}{A}_{22}^{?1}{A}_{21}$ is called the Schur complement of $A_{22}$ in $A$, denoted by $S_{\text{ch?}}\left(A_{22}\right)$.

Lemma 1. Let $\stackrel{eq}{=}$ represent the equivalence relation between two matrices. Then for the partitioned matrix $A$ the following conclusions hold.

When $A_{11}$ is nonsingular, $$A\stackrel{eq}{=}\left[\begin{array}{cc}{A}_{11}& 0\\ 0& A_{22}?A_{21}{A}_{11}^{?1}{A}_{12}\end{array}\right]=\left[\begin{array}{cc}{A}_{11}& 0\\ 0& S_{ch}\left(A_{11}\right)\end{array}\right]$$ and hence $A$ is nonsingular if and only if $S_{ch}\left(A_{11}\right)$ is nonsingular, and $$\det A=\det A_{11}\det S_{ch}\left(A_{11}\right)$$

When $A_{22}$ is nonsingular, $$A\stackrel{eq}{=}\left[\begin{array}{cc}{A}_{11}?A_{12}{A}_{22}^{?1}{A}_{21}& 0\\ 0& A_{22}\end{array}\right]=\left[\begin{array}{cc}{S}_{ch}\left(A_{22}\right)& 0\\ 0& A_{22}\end{array}\right]$$ hence $A$ is nonsingular if and only if $S_{ch}\left(A_{22}\right)$ is nonsingular, and $\det A=\det A_{22}\det S_{ch}\left(A_{22}\right)$.
note: refer to https://blog.csdn.net/weixin_44382195/article/details/102991813 for the definition of the equivalence relation.

Lemma 2. Given the matrices $A_{11}=A_{11}^T,A_{22}=A_{22}^T$ and $A_{12}$ with appropriate dimensions. The following LMIs are equivalent:

$\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{12}^T& A_{22}\end{array}\right]?0$

$A_{22}=A_{22}^T?0;A_{11}?A_{12}{A}_{22}^{?1}{A}_{12}^T?0$

$A_{11}=A_{11}^T?0;A_{22}?A_{12}^T{A}_{11}^{?1}{A}_{12}?0$.

Lemma 3. Given the matrices $A=A^T,C=C^T$ and $B$ with appropriate dimensions. The following LMIs are equivalent:

$\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{12}^T& A_{22}\end{array}\right]?0$

$A_{22}=A_{22}^T?0;A_{11}?A_{12}{A}_{22}^{?1}{A}_{12}^T?0$

$A_{11}=A_{11}^T?0;A_{22}?A_{12}^T{A}_{11}^{?1}{A}_{12}?0$.

與50位技術專家面對面20年技術見證，附贈技術全景圖

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