Lord it's been a while since Linear Algebra. IIRC you multiply your source matrix by the inverse of the transformation matrix to restore the source to it's original state. Edit: ahh, here you go ...

Richard Sinkhorn, Linear Transformations Under Which the Doubly Stochastic Matrices are Invariant, Proceedings of the American Mathematical Society, Vol. 27, No. 2 (Feb., 1971), pp. 213-221 ...

Linear Algebra Vector spaces, linear transformation, matrix representation, inner product spaces, isometries, least squares, generalised inverse, eigen theory, quadratic forms, norms, numerical ...

Transforming light: illustration of how an arbitrary linear transform can be achieved in an all-optical system using diffractive surfaces (Courtesy: Ozcan Lab/UCLA) Researchers in the US have shown ...

Two linear models with error structure of the nested type are considered. Transformations are presented by which uncorrelated errors with constant variances are ...