Given the inverse of a block matrix $X^{-1}$, where
$$ X=\left(\begin{array}{cc} A & B \end{array}\right). $$
A is $m\times n$ and B is $m\times(n-m)$.
Can I obtain the pseudo-inverse of A from $X^{-1}$?
2026-03-27 21:35:34.1774647334
Given the inverse of a block matrix...
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We note that $$ I_n = X^{-1}X = X^{-1}\pmatrix{A&B} = \pmatrix{X^{-1}A & X^{-1}B} $$ Thus, we have $$ X^{-1}A = \pmatrix{ 1&&0\\ &\ddots&\\ &&1\\ &0&\\ && } $$ Where the matrix on the right is $m \times n$. If we take the first $m$ rows of $X^{-1}$ and call the resulting matrix $M$, we find that $$ MA = \pmatrix{ 1&&0\\ &\ddots&\\ 0&&1\\ } $$ Which means that $M$ is the left-sided pseudo inverse, depending on your definition.