  Multivariate multiple linear regression (MMLR), which occurs in a number of
practical applications, generalizes traditional least squares (multivariate
linear regression) to multiple right-hand sides. We extend recent MLR analyses
to sketched MMLR in general Schatten $p$-norms by interpreting the sketched
problem as a multiplicative perturbation. Our work represents an extension of
Maher's results on Schatten $p$-norms. We derive expressions for the exact and
perturbed solutions in terms of projectors for easy geometric interpretation.
We also present a geometric interpretation of the action of the sketching
matrix in terms of relevant subspaces. We show that a key term in assessing the
accuracy of the sketched MMLR solution can be viewed as a tangent of a largest
principal angle between subspaces under some assumptions. Our results enable
additional interpretation of the difference between an orthogonal and oblique
projector with the same range.
