# Inner product of columns of a matrix

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

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.
• @M.FarrokhiD.G. The matrix of inner products is $A A^t,$ so its determinant is the square of the determinant of $A.$ For various other formulas, check out my paper Surface Area and Other Measures of Ellipsoids. – Igor Rivin Jan 26 '17 at 4:45