Thread: 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?

I think you have it.

For column vectors ,

Just mention that each entry of the product is , so { 1 if i=j and 0 if i =/ j } by orthogonality.

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

I could be wrong but I think it is there.

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

Originally Posted by snowtea

I think you have it.

For column vectors ,

Just mention that each entry of the product is , 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...

Is anyone able to give me the full proof?

Iff. two ways.

1 direction


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