The algebraic polar decomposition is defined as: $$A=QS$$ Where $Q$ is a matrix with orthonormal columns and $S$ is a symmetric matrix. The question is, when is such a factorization possible? I started with the SVD of $A$:
$$A = U \Sigma V^T$$ Here, $U$ and $V$ are orthonormal matrices and $\Sigma$ is diagonal matrix with the singular values of $A$.
This leads to: $$S = V (\Sigma \Sigma)^.5 V^T$$ and $$QV(\Sigma \Sigma)^.5 = U\Sigma$$
If $A$ is full column rank, then $(\Sigma \Sigma)^.5$ will be invertible and we can easily solve for $Q$. The interesting case is when $A$ is not full column rank. Here are some cases -
1) When $A$ has more columns than rows (fat matrix), this is clearly impossible since we are hoping to get a $Q$ with orthonormal columns that span a space higher than the size of the column space itself.
2) When $A$ is square but not singular, here is an example where it is possible:
In : a Out: array([[ 1. , 0.5, 0.4], [ 0.5, 1. , 0. ], [ 0. , 0. , 0. ]]) In : s Out: array([[ 0.92417515, 0.52002785, 0.35421932], [ 0.52002785, 0.9896415 , 0.01344378], [ 0.35421932, 0.01344378, 0.18533196]]) In : q Out: array([[ 0.88554831, 0.03360946, 0.46332991], [ 0.07725369, 0.97283677, -0.21822117], [-0.45807867, 0.22903933, 0.8588975 ]])
So, my question is - when is a decomposition like this possible and when not?