<s>[INST] Correlation between two matrices what is the correlation/relationship between two matrices
A = \begin{bmatrix}4&0&0&0\\0&3&0&0\\0&0&2&0\end{bmatrix}
B = \begin{bmatrix}1/4&0&0\\0&1/3&0\\0&0&1/2\\0&0&0\end{bmatrix}
where A*B = \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}
A cannot be an inverse of B because it's not square. So is there a word that describes the relation between this two matrices A and B?
EDIT1: Changed dimensions of A from 4X4 to 3X4
 [/INST] You may be reaching for the Moore-Penrose pseudoinverse, usually denoted as $A^+$, which is also defined for rectangular and singular matrices.
In this case:
$$B^+ = \begin{bmatrix}1/4&0&0\\0&1/3&0\\0&0&1/2\\0&0&0\end{bmatrix}^+
= \begin{bmatrix}4&0&0&0\\0&3&0&0\\0&0&2&0\end{bmatrix}$$
And:
$$B^+\cdot B=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$$
$$B\cdot B^+=\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&0\end{bmatrix}$$
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