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Let $A=BD^{\dagger}B^{T}$. I am looking for conditions under which $A^{\dagger}$ is a "nice" expression in $B$ and $D$ and their Moore-Penrose pseudo-inverses.

Do you know of such conditions?

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Felix, a reverse order law for the triple product might give some nice answer. For instance, see R. Hartwig, The reverse order law revisited, Linear Algebra Appl., 76 (1986), 241-246. – user27020 Nov 15 '12 at 3:10
    
@SJ: Thanks a lot, this looks very relevant. Maybe you would like to write this as an answer so I can accept it? – Felix Goldberg Nov 15 '12 at 8:17

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