How to find matrix $A$ from the relation: $A\times (A^TA)^{-1}\times A^T = B$

Question

Kindly help me in the following:

• I have two Matrices, $A$ of size $(n\times m)$; and $B$ of size $(n\times n)$, where $n>m$.
• $A$ is unknown, but $B$ is known.
• $(A^TA)$ is invertible
• $B$ is symmetric and invertible.
• $B$ is not an Identity Matrix.

• The two Matrices are related to each other as follows:

  $A\times (A^TA)^{-1}\times A^T = B$

Is there a closed-form solution to finding $A$ from $B$ ?

Thank you in advance!

Answer 1

Not hard to see

1) $BA=A$

2) $\mathrm{rank}A=m$

So all $m$ columns of matrix $A$ are linearly independent eigenvectors which eigenvalues are equal to 1. The set of all eigenvectors with eigenvalues=1 is a space of dimension $k$, $m\le k<n$ (since $B\ne I$ then $k\ne n$) Let $C_1,...,C_k$ be a basis of this space, and $C=(C_1,...,C_k)$ be the $n\times k$ matrix. So, the set of answers (appropriate matrix $A$) is $$A=CD,$$ where $D$ is any $k\times m$ matrix of the rank $m$.