I am trying to solve the matrix equation $A = B^TC$ for $C$, where $A$, $B$, and $C$ are all non-square matrices. I know that I need to utilize $M^TM$ in order to take the inverse. I'm just not sure how to isolate $C$ in the equation provided.
2026-04-09 00:50:29.1775695829
Using inverse of transpose matrix to cancel out terms?
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We have the matrix equation in $\mathrm C$
$$\mathrm B^\top \mathrm C = \mathrm A$$
Let's left-multiply both sides by $\mathrm B$
$$\mathrm B \mathrm B^\top \mathrm C = \mathrm B \mathrm A$$
If $\mathrm B$ has full row rank, then $\mathrm B \mathrm B^\top$ is invertible. Hence,
$$\mathrm C = (\mathrm B \mathrm B^\top)^{-1} \mathrm B \mathrm A$$
If $\mathrm B$ does not have full row rank, then we vectorize $\mathrm B^\top \mathrm C = \mathrm A$, which gives us the linear system
$$(\mathrm I \otimes \mathrm B^\top) \operatorname{vec} (\mathrm C) = \operatorname{vec} (\mathrm A)$$
which can be solved using Gaussian elimination.