# None element of orthogonal matrix can't have unit modulus larger then 1

None element of orthogonal matrix can't have unit modulus larger then 1.

I've tried to use the properties of orthogonal matrices ( $|det(A)| = 1$ and $Q^T=Q^{-1}$ ) but I couldn't find out how they could help me.

• Do you know that the columns of an orthogonal $n \times n$ matrix form a set of orthonormal vectors in $\mathbf{R}^{n}$ (or can you see how to deduce this easily from $Q^{T}Q = I$)...? – Andrew D. Hwang Jun 16 '17 at 17:11
• @AndrewD.Hwang I know that,but how can that help me? – Dragan Zrilić Jun 16 '17 at 17:16
• The magnitude squared of a vector is the sum of the squares of its components; for a unit vector, the magnitude squared is $1$. ;) – Andrew D. Hwang Jun 16 '17 at 18:07

Note that $$Q^T Q =1$$. So, given $$q_i$$ the row $$i$$ of Q we have $$q_i^Tq_i=1$$. In other words, for all k, $$q_{ik}^2 \leq \sum q_{ij}^2 = 1$$. It means that $$|q_{ik}|\leq 1$$.