I've got a question concerning semi-orthogonal matrices.
In their book 'Matrix Algebra', Abadir and Magnus define a semi-orthogonal matrix as a matrix A satisfying one of the two equations: $A^T\cdot A=1$ or $A \cdot A^T= 1$, but not necessarily both.
In the Wikipedia article on semi-orthogonal matrices, the isometry property of transformations with semi-orthogonal matrices (semi-unitary transformations) is mentioned:
$||Ax||_2=||x||_2$ for all $x$ in $R^n$
which is true if $A^T\cdot A=I$ (A has more columns than rows), but $A\cdot A^T \neq I$ (A has more rows than columns).
My question is very simple: Is this true? I love Wikipedia, but now I'm at a point where I need more "evidence" to back this up. Does any one of you know any literature about this, or a way I can show that the isometry property of semi-orthogonal matrices is valid?
I would really appreciate your help!
Cheers,
bruschino
Best Answer
You have $$\Vert Ax \Vert_2=\langle Ax,Ax \rangle = \langle x,A^TAx \rangle = \langle x,x \rangle = \Vert x \Vert_2$$