What information about a (square) matrix we earn if the inner product of its columns are known?



Inner product of columns of a matrix
You have determined $A^\*A$, or, alternatively, you know $A$ up to pre-multiplication by a unitary matrix $U$. So you know the $R$ factor of its QR factorization, and the factors $\Sigma$ and $V$ of its SVD. In particular, among other things, you know singular values and right singular vectors, the sign of its determinant, its rank, its kernel, its Euclidean and Frobenius norm.


