Middle inverse. Suppose A is an n×p matrix and B is a q×n matrix. If a p×q matrix X exists that satisfies AXB = I, we call it a middle inverse of the pair A, B. (This is not a standard concept.) Note that when A or B is an identity matrix, the middle inverse reduces to the right or left inverse, respectively.
(a) Describe the conditions on A and B under which a middle inverse X exists. Give your answer using only the following four concepts: Linear independence of the rows or columns of A, and linear independence of the rows or columns of B. You must justify your answer.
(b) Give an expression for a middle inverse, assuming the conditions in part (a) hold.
My attempt for part A:
(a) In order for X to exist, A and B have to be the non-zero matrix. The rows of A must be linearly independent and the columns of B must be linearly independent.
Reasons:
AXB = I so A(XB) = I, so XB will be the right inverse of A, so the rows of A are linearly independent.
AXB = I so (AX)B = I, so AX will be the left inverse of B, so the columns of B are linearly independent.
(a)
In order for X to exist, A and B have to be the non-zero matrix.
AXB = I so A(XB) = I, so XB will be the right inverse of A, so the rows of A are linearly independent.
AXB = I so (AX)B = I, so AX will be the left inverse of B, so the columns of B are linearly independent.
(b)
A has a right inverse, so $AA^* = I$
B has a left inverse, so $B^*B = I$
so
$AXB=I$
$AA^*B^*B = I$
$X = A^*B^*$
By Pseudo Inverse:
$A^* = A^T(AA^T)^{-1}$
$B^* = (B^TB)^{-1}B^T$
so
$X = A^*B^* = A^T(AA^T)^{-1}(B^TB)^{-1}B^T $