Suppose I have a device that can compute 2x2 (complex) matrix inverses. (For now, assume only invertible matrices, $A$, are ever provided as input):

$A\triangleq \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}$

$A^{-1}= \frac{1}{(a_{11}a_{22}-a_{12}a_{21})}\begin{bmatrix} a_{22} & -a_{12}\\ -a_{21} & a_{11}\\ \end{bmatrix}$

I can use the same device to compute the pseudoinverse of a 2x1 vector (where $^H$ denotes complex conjugate transpose):


For example, I can achieve this by inputting the following matrix to my matrix inversion device (where $^*$ denotes complex conjugate):

$B= \begin{bmatrix} b_{1} & -b^*_2\\ b_{2} & b^*_1\\ \end{bmatrix}$

Since the inverse is:

$B^{-1}= \frac{1}{(\underline{b}^H\underline{b})}\begin{bmatrix} b^*_1 & b^*_2\\ -b_2 & b_1\\ \end{bmatrix}$

and the first row of the result is $\underline{b}^+$.

My question is whether this is generalisable to larger matrices. For example, if I have a 3x3 matrix inversion device, can I arrange inputs such that I can read off a 3x2 matrix pseudoinverse? Or have I just stumbled upon the only special case?


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