# Math Help - Orthogonal Matrix proof

1. ## Orthogonal Matrix proof

An n x n matrix Q is said to be an orthogonal matrix if the column vectors of Q form an orthornormal minimal spanning set of R^n. Prove the following theorem.

An n x n matrix Q is orthogonal if and only if Q^TQ = In (When Q^T is the transpose of Q)

I'm struggling with this.

Q = (Q1, Q2, ..., Qn) so Q^T = Row vector of Q.

So Q^TQ = Is an n x n matrix where the ij components { 1 if i=j or 0 if i =/ j }

So QTQ = In

If you can understand what I put, I kind of proved one way. Could someone help with the full proof?

2. I think you have it.

For column vectors $v_1, v_2, ...$,

Just mention that each entry of the product is $(Q^TQ)_{ij} = v_i \cdot v_j$, so { 1 if i=j and 0 if i =/ j } by orthogonality.

3. I am almost positive I have this proven in the Sticky in this forum.

I could be wrong but I think it is there.

4. Originally Posted by dwsmith
I am almost positive I have this proven in the Sticky in this forum.

I could be wrong but I think it is there.
I searched but could not find it

5. Originally Posted by snowtea
I think you have it.

For column vectors $v_1, v_2, ...$,

Just mention that each entry of the product is $(Q^TQ)_{ij} = v_i \cdot v_j$, so { 1 if i=j and 0 if i =/ j } by orthogonality.
I haven't though, as it is an if and only if statement, and I've only shown it one way...

6. Is anyone able to give me the full proof?

7. Iff. two ways.

1 direction
$P\Rightarrow Q$

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