2021-05-19 Schur补引理及证明

Definition 1. Consider the partitioned matrix
A=[A11A12A21A22]A=\left[\begin{array}{ll} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right] A=[A11A21A12A22]

When A11A_{11}A11 is nonsingular, A22−A21A11−1A12A_{22}-A_{21} A_{11}^{-1} A_{12}A22−A21A11−1A12 is called the Schur complement of A11A_{11}A11 in AAA, denoted by Sch(A11)S_{c h}\left(A_{11}\right)Sch(A11).

When A22A_{22}A22 is nonsingular, A11−A12A22−1A21A_{11}-A_{12} A_{22}^{-1} A_{21}A11−A12A22−1A21 is called the Schur complement of A22A_{22}A22 in AAA, denoted by Sch (A22)S_{\text {ch }}\left(A_{22}\right)Sch (A22).

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

When A11A_{11}A11 is nonsingular, A=eq[A1100A22−A21A11−1A12]=[A1100Sch(A11)]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] A=eq[A1100A22−A21A11−1A12]=[A1100Sch(A11)] and hence AAA is nonsingular if and only if Sch(A11)S_{c h}\left(A_{11}\right)Sch(A11) is nonsingular, and det⁡A=det⁡A11det⁡Sch(A11)\operatorname{det} A=\operatorname{det} A_{11}\operatorname{det} S_{c h}\left(A_{11}\right) detA=detA11detSch(A11)

When A22A_{22}A22 is nonsingular, A=eq[A11−A12A22−1A2100A22]=[Sch(A22)00A22]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_{c h}\left(A_{22}\right) & 0 \\ 0 & A_{22} \end{array}\right] A=eq[A11−A12A22−1A2100A22]=[Sch(A22)00A22] hence AAA is nonsingular if and only if Sch(A22)S_{c h}\left(A_{22}\right)Sch(A22) is nonsingular, and det⁡A=det⁡A22det⁡Sch(A22)\operatorname{det} A=\operatorname{det} A_{22}\operatorname{det} S_{c h}\left(A_{22}\right)detA=detA22detSch(A22).
note: refer to https://blog.csdn.net/weixin_44382195/article/details/102991813 for the definition of the equivalence relation.

Lemma 2. Given the matrices A11=A11T,A22=A22TA_{11}=A_{11}^{T}, A_{22}=A_{22}^{T}A11=A11T,A22=A22T and A12A_{12}A12 with appropriate dimensions. The following LMIs are equivalent:

[A11A12A12TA22]≻0\left[\begin{array}{cc}A_{11} & A_{12} \\ A_{12}^{T} & A_{22}\end{array}\right]\succ0[A11A12TA12A22]≻0

A22=A22T≻0;A11−A12A22−1A12T≻0A_{22}=A_{22}^{T}\succ0 ; A_{11}-A_{12} A_{22}^{-1} A_{12}^{T}\succ0A22=A22T≻0;A11−A12A22−1A12T≻0

A11=A11T≻0;A22−A12TA11−1A12≻0A_{11}=A_{11}^{T}\succ0 ; A_{22}-A_{12}^{T} A_{11}^{-1} A_{12}\succ0A11=A11T≻0;A22−A12TA11−1A12≻0.

Lemma 3. Given the matrices A=AT,C=CTA=A^{T}, C=C^{T}A=AT,C=CT and BBB with appropriate dimensions. The following LMIs are equivalent:

[A11A12A12TA22]≺0\left[\begin{array}{cc}A_{11} & A_{12} \\ A_{12}^{T} & A_{22}\end{array}\right]\prec0[A11A12TA12A22]≺0

A22=A22T≺0;A11−A12A22−1A12T≺0A_{22}=A_{22}^{T}\prec0 ; A_{11}-A_{12} A_{22}^{-1} A_{12}^{T}\prec0A22=A22T≺0;A11−A12A22−1A12T≺0

A11=A11T≺0;A22−A12TA11−1A12≺0A_{11}=A_{11}^{T}\prec0 ; A_{22}-A_{12}^{T} A_{11}^{-1} A_{12}\prec0A11=A11T≺0;A22−A12TA11−1A12≺0.

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