Originally Posted by

**HallsofIvy** No, Tonio. M is not orthogonal. Two equivalent definitions of "orthogonal matrix" are:

1) It's columns are orthogonal vectors.

That is not true for your matrix, M, because the dot products of the first and third columns is 1, not 0.

2) The transpose equals the inverse matrix.

That is not true for your matrix, M, because

$\displaystyle \begin{pmatrix}1 & 0 & 0 \\0 & -1 & 0 \\ 1 & 0 & 1\end{pmatrix}\begin{pmatrix}1 & 0 & 1 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{pmatrix}= \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 2\end{pmatrix}$.