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


1 Answer 1


You have $$\Vert Ax \Vert_2=\langle Ax,Ax \rangle = \langle x,A^TAx \rangle = \langle x,x \rangle = \Vert x \Vert_2$$


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