The assignment:
a) Prove that square-matrix A is orthogonal if and only if A has orthonormal columns.
b) Prove that square-matrix A is orthogonal if and only if A has orthonormal rows.
So I know that A matrix has orthonormal columns if and if only $A^TA=I$.
But how about orthonormal rows? Should I use $AA^T=I$ ?
b) For example, can I prove like this (?) :
Let be $A=\begin{bmatrix} a_1 & a_2 & a_3 \end{bmatrix}$
$AA^T=I$
$AA^T=\begin{bmatrix} a_1 & a_2 & a_3 \end{bmatrix} \times \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix}^T = \begin{bmatrix} a_1a_1^T & a_2a_2^T & a_3a_3^T \end{bmatrix}$
$a_1a_1^T=1 \quad\quad a_2a_2^T=1 \quad\quad a_3a_3^T=1$
So $A$ is orthogonal, because rows of matrix A are orthonormal. $\Box$
a) I did it like this which I think is correct:
$A^TA=I$
$A^TA=\begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix}^T \times \begin{bmatrix} a_1 & a_2 & a_3 \end{bmatrix} = \begin{bmatrix} a_1^Ta_1 & a_1^Ta_2 & a_1^Ta_3 \\ a_2^Ta_1 & a_2^Ta_2 & a_2^Ta_3 \\ a_3^Ta_1 & a_3^Ta_2 & a_3^Ta_3 \end{bmatrix}$
$a_1^Ta_2=0 \quad a_1^Ta_3=0$
$a_2^Ta_1=0 \quad a_2^Ta_3=0$
$a_3^Ta_1=0 \quad a_3^Ta_2=0$
$a_1^Ta_1=1 \quad\quad a_2^Ta_2=1 \quad\quad a_3^Ta_3=1$
So $A$ is orthogonal, because columns of matrix A are orthonormal. $\Box$