Question

Are the rows of an orthogonal matrix A necessarily orthonormal?

Short answer

Yes.

Step-by-step solution

Definition of an Orthonormal matrix.

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.

One way to express this is. Where QT is the transpose of Q and I is the identity matrix.

For example,

$Q^{T}Q\left(\begin{array}{c}{{\stackrel{\rightharpoonup }{q}}_1}^T\\ {{\stackrel{\rightharpoonup }{q}}_2}^T\\ {{\stackrel{\rightharpoonup }{q}}_3}^T\\ .\\ .\\ {{\stackrel{\rightharpoonup }{q}}_n}^T\end{array}\right)\left({\stackrel{\rightharpoonup }{q}}_1,{\stackrel{\rightharpoonup }{q}}_2,{\stackrel{\rightharpoonup }{q}}_3,........{\stackrel{\rightharpoonup }{q}}_n\right)$

According to above definition.

Let A be an orthogonal matrix. Let $A={\stackrel{\rightharpoonup }{v}}_1....{\stackrel{\rightharpoonup }{v}}_n$.

Observe that for any $\stackrel{\rightharpoonup }{v}\in R^n$,

$$\begin{gathered} ||A^T\left(\stackrel{\rightharpoonup }{v}\right)||=||A\left(A^T\left(\stackrel{\rightharpoonup }{v}\right)\right)|| \\ =||I\stackrel{\rightharpoonup }{v}|| \\ =||\stackrel{\rightharpoonup }{v}|| \end{gathered}$$

Hence, the row of an orthogonal matrix is orthonormal.