1. The problem statement, all variables and given/known data I want to show that for an n x n matrix A with complex entries, if [itex]\left\|Ax\right\|=\left\|x\right\|[/itex] for any vector x in C^n, then the rows of A are an orthonormal basis of C^n. 2. Relevant equations 3. The attempt at a solution All I've managed to do so far is show that the columns of A all have length 1, which you can get by taking x to be e1. Is there a strategy of showing this for the rows and for showing orthogonality without having to write out the terms of the matrix multiplication Ax (I tried this and quickly became bogged down in notation)?