I am trying to prove that AA+vec(a)_i=vec(a)_i, AA i=1..n is equal to the more popular version AA^+A=A

Genesis Gibbs 2022-09-06 Answered
I am trying to prove that A A + a i = a i , i = 1.. n is equal to the more popular version A A + A = A. I started by setting up a system of equations as follows:
{ A A + a 1 = a 1 A A + a n = a n
Then I turned it into a matrix form:
[ A A + a 1 A A + a n ] = [ a 1 a n ]
A A + [ a 1 a n ] = [ a 1 a n ]
Since a i is a column vector it has shape (m x 1) which seems to produce weird vector in the last equation. Is this part of a "proof" any good or is it a bad idea from the begining?
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Answers (1)

Paige Paul
Answered 2022-09-07 Author has 11 answers
I suppose a j means the j-th column of A. Then A is equal to [ a 1 a n ] . So, the correct identity should be
[ A A + a 1 A A + a n ] = A A + [ a 1 a n ] = A A + A = A = [ a 1 a n ] .
The matrix-vector product A A + [ a 1 a n ] does not make sense because the size of A A + (which is, for example, n × n when A is n × n) does not match the size of the the column vector [ a 1 a n ] (which is n 2 × 1 in this example).
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