线性方程组 A x = b \bold{Ax=b} Ax=b或 A X = B \bold{AX=B} AX=B型方程有解定理 R ( A ) ⩽ R ( A , B ) R(\bold{A})\leqslant{R(\bold{A,B})} R(A)⩽R(A,B)
等价矩阵同秩
转置矩阵同秩
秩的定义
公式
R ( k A ) = R ( A ) R(k\bold{A})=R(\bold{A}) R(kA)=R(A), ( k ≠ 0 ) (k\neq{0}) (k=0)
R ( A ) R(\bold{A}) R(A)= R ( A T ) R(\bold{A^{T}}) R(AT)
R ( A B ) ⩽ min ( R ( A ) , R ( B ) ) R(\bold {AB})\leqslant{\min(R(\bold A),R(\bold B))} R(AB)⩽min(R(A),R(B))
max ( R ( A ) , R ( B ) ) ⩽ R ( A , B ) ⩽ R ( A ) + R ( B ) \max(R(\bold{A}),R(\bold{B}))\leqslant{R(\bold{A,B})}\leqslant{R(\bold{A})+R(\bold{B})} max(R(A),R(B))⩽R(A,B)⩽R(A)+R(B)
max ( R ( A ) , R ( B ) ) ⩽ R ( A B ) ⩽ R ( A ) + R ( B ) \max(R(\bold{A}),R(\bold{B}))\leqslant{R\begin{pmatrix}\bold A\\\bold B\end{pmatrix}}\leqslant{R(\bold{A})+R(\bold{B})} max(R(A),R(B))⩽R(AB)⩽R(A)+R(B)
R ( A + B ) ⩽ R ( A + B B ) = R ( A B ) ⩽ R ( A ) + R ( B ) R(\bold{\bold{A+B}}) \leqslant{R\begin{pmatrix}\bold{A+B}\\\bold B\end{pmatrix}} ={R\begin{pmatrix}\bold A\\\bold B\end{pmatrix}} \leqslant{R(\bold A)+R(\bold B)} R(A+B)⩽R(A+BB)=R(AB)⩽R(A)+R(B)
R ( A T A ) ⩽ R ( A ) R(\bold{A^{T}{A}})\leqslant{R(\bold A)} R(ATA)⩽R(A)
R ( A 0 0 B ) R\begin{pmatrix}\bold{A}&\bold{0}\\\bold{0}&\bold{B}\end{pmatrix} R(A00B)= R ( A ) + R ( B ) R(\bold{A})+R(\bold{B}) R(A)+R(B)
若 ∣ A ∣ ≠ 0 |\bold{A}|\neq{0} ∣A∣=0,则 R ( A B ) R(\bold{AB}) R(AB)= R ( B A ) = R ( B ) R(\bold{BA})=R(\bold{B}) R(BA)=R(B)
若 A m × n B n × l = 0 \bold{A}_{m\times{n}}\bold{B}_{n\times{l}}=\bold{0} Am×nBn×l=0,则 R ( A ) + R ( B ) ⩽ n R(\bold A)+R(\bold B)\leqslant{n} R(A)+R(B)⩽n
