Useful lemmas Proofs involving the Moore–Penrose inverse
1 useful lemmas
1.1 lemma 1: a*a = 0 ⇒ = 0
1.2 lemma 2: a*ab = 0 ⇒ ab = 0
1.3 lemma 3: abb* = 0 ⇒ ab = 0
useful lemmas
these results used in proofs below. in following lemmas, matrix complex elements , n columns, b matrix complex elements , n rows.
lemma 1: a*a = 0 ⇒ = 0
the assumption says elements of a*a zero. therefore,
0
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2
{\displaystyle 0=\operatorname {tr} (a^{*}a)=\sum _{j=1}^{n}(a^{*}a)_{jj}=\sum _{j=1}^{n}\sum _{i=1}^{m}(a^{*})_{ji}a_{ij}=\sum _{i=1}^{m}\sum _{j=1}^{n}|a_{ij}|^{2}}
.
therefore,
a
i
j
{\displaystyle a_{ij}}
equal 0 i.e.
a
=
0
{\displaystyle a=0}
.
lemma 2: a*ab = 0 ⇒ ab = 0
0
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∗
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b
⇒
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⇒
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⇒
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b
(
lemma 1
)
{\displaystyle {\begin{alignedat}{3}&0\,&&=a^{*}ab&&&\\\rightarrow \,&0\,&&=b^{*}a^{*}ab&&&\\\rightarrow \,&0\,&&=(ab)^{*}(ab)&&&\\\rightarrow \,&0\,&&=ab&&&({\text{by lemma 1}})\end{alignedat}}}
lemma 3: abb* = 0 ⇒ ab = 0
this proved in manner similar argument of lemma 2 (or taking hermitian conjugate).
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