This is probably an elementary question.
Suppose that the column vectors of a matrix (in $\mathbb R^{d\times d}$) are all unit vectors. That implies that its determinant is bounded in absolute value by one.
If the determinant of that matrix is one, is the matrix orthogonal (and therefore all column vectors are in fact orthogonal to each other)?
To cross t's and dot i's : the norm is the Euclidean norm, and the scalar product is the usual scalar product in $\mathbb R^d$.
Because of the volume interpretation of the determinant, the answer should be yes. But what is a simple proof?
Best Answer
Yes, it follows from the so called Hadamard's inequality. To put it simple, determinant is equal to volume of the cube spanned by columns. Since in your case columns are normalized and volume is 1 they must be orthogonal.