I need to implement some numpy matrix operations but very frequently the code breaks because a matrix determinant tends to be $0$ and the inverse can not be calculated. Is there a way to calculate the value of $e$ in the following equation assuming we know $T$ and $s$ and they are a MxM matrix and Mx1 vector, respectively. $T=se^T-es^T$
PS: This has to do with fusing classifier predictions, and $\hat{T}=\hat{s}e^T-e\hat{s}^T$ will be a matrix that results after a singular value thresholding operation on a the mean matrix $T$ of the prediction scores. In order to recreate the fused prediction $\hat{s}$ i need to know $e$.