擬似逆行列
擬似逆行列は以下のような性質を持つ。
( A + ) + = A {\displaystyle (A^{+})^{+}=A}
( A ⊺ ) + = ( A + ) ⊺ , ( A A ⊺ ) + = ( A + ) ⊺ A + {\displaystyle (A^{\intercal })^{+}=(A^{+})^{\intercal },(AA^{\intercal })^{+}=(A^{+})^{\intercal }A^{+}}
rank  A = rank  B = n ⟹ ( A B ) + = B + A + {\displaystyle \operatorname {rank} A=\operatorname {rank} B=n\implies (AB)^{+}=B^{+}A^{+}}
A + = ( A ⊺ A ) + A ⊺ = A ⊺ ( A A ⊺ ) + {\displaystyle A^{+}=(A^{\intercal }A)^{+}A^{\intercal }=A^{\intercal }(AA^{\intercal })^{+}}
m × n {\displaystyle m\times n} 行列 A に対して rank  A = m ⟹ A + = A ⊺ ( A A ⊺ ) − 1 rank  A = n ⟹ A + = ( A ⊺ A ) − 1 A ⊺ {\displaystyle {\begin{aligned}\operatorname {rank} A=m&\implies A^{+}=A^{\intercal }(AA^{\intercal })^{-1}\\\operatorname {rank} A=n&\implies A^{+}=(A^{\intercal }A)^{-1}A^{\intercal }\end{aligned}}}