if $Q^TQ = I$, can we get $QQ^T=I$

Question

if we only know $Q^TQ = I$, can we get $QQ^T=I$?

where $I$ is the identify matrix, $Q \in R^{m \times m}$

Obviously... There is a case where this is called 'generalized inverse'. — Anik Bhowmick, Oct 01 '18 at 18:40
If a matrix is invertible, its inverse is unique and $Q^{-1} Q = QQ^{-1} = I$ The set-up allows us to say $Q^T = Q^{-1}$ — Doug M, Oct 01 '18 at 18:46
See: https://math.stackexchange.com/questions/3852/if-ab-i-then-ba-i — yurnero, Oct 01 '18 at 18:50

Mark · Accepted Answer · 2018-10-01T18:52:50.410

4

Let's say $Q$ is a matrix of size $m\times n$. Then $Q^TQ=I$ implies $QQ^T=I$ iff $Q$ is a square matrix. If $Q$ is a square matrix then I assume you know $Q^TQ=I$ implies $Q$ is invertible and $Q^T=Q^{-1}$. Hence $I=QQ^{-1}=QQ^T$. As for the other direction suppose $Q^TQ=I_n$ and $QQ^T=I_m$. You can check that for any two matrices $A$ and $B$ (not necessary square!) that can be multiplied from both sides $tr(AB)=tr(BA)$. So that way we get:

$m=tr(I_m)=tr(QQ^T)=tr(Q^TQ)=tr(I_n)=n$

So $m=n$ and $Q$ is a square matrix.

edited Oct 01 '18 at 18:52

answered Oct 01 '18 at 18:46

Mark

39,605

Simple and nice use of $\mathop{\rm tr}$. – GNUSupporter 8964民主女神地下教會 Oct 01 '18 at 18:52

score 1 · Answer 2 · answered Oct 01 '18 at 18:43

1

Yes in the case of square matrices. If $Q^TQ = I$ then $Q^{-1} = Q^T$ so $QQ^T = QQ^{-1} = I$.

Essentially this works because for square matrices a left hand inverse is a right hand inverse.

answered Oct 01 '18 at 18:43

Mohit

1,154

score 0 · Answer 3 · answered Oct 01 '18 at 19:21

0

Yes. If $Q$ is orthogonal matrix i.e. $Q^TQ = I$ then it also follows that $ Q^T = Q^{-1}$. Putting it together we have:

$$\begin{align} Q^TQ = I = QQ^{-1} = QQ^T \end{align}$$

answered Oct 01 '18 at 19:21

minggli

25

if $Q^TQ = I$, can we get $QQ^T=I$

3 Answers3