Is the definition of an orthogonal matrix: 1. a matrix where all rows are orthonormal AND all columns are orthonormal OR 2. a matrix where all rows are orthonormal OR all columns are orthonormal? On my textbook it said it is AND (case 1), but if that is true, there's a problem: Say we have a square matrix A, and we find its eigenvectors, they are all distinct so A is diagonalizable. We put the normalized eigenvectors of A as the columns of a matrix P, and (our prof told us) P becomes orthogonal and P^-1 = P^T. My question is how did P become orthogonal straight away? By only normalizing its columns how did we guarantee that its rows are also orthonormal?