If G is a generalized inverse of a matrix A (i.e. AGA=A), then is it true that every generalized inverse of A can be written in the form G+B−GABAG for some matrix B of same order as G?

Jamar Hays 2022-09-13 Answered
If G is a generalized inverse of a matrix A (i.e. AGA=A), then is it true that every generalized inverse of A can be written in the form G+B−GABAG for some matrix B of same order as G?
I will show that this matrix is a generalized inverse for every matrix B, since
A ( G + B G A B A G ) A = A G A + A B A A G A B A G A = A + A B A A B A = A
But I cant conclude that every generalized inverse of A can be written in this form. Help with that
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Answers (1)

Saige Barton
Answered 2022-09-14 Author has 15 answers
Let f ( X ) = A X A , g ( X ) = G A X A G and π = id g. One can verify that g and in turn π are idempotent. Also, f π = 0. Therefore
range ( π ) ker ( f ) ker ( g ) .
However, as π is g are complementary projections to each other, we have range ( π ) = ker ( g ). Thus range ( π ) = ker ( f ) = ker ( g ) by the sandwich principle.
Now, for any generalised inverse X of A, we have X G ker ( f ) = range ( π ). Hence X = G + π ( B ) for some matrix B.
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