Gram-Schmidt verifying orthonormal basis

Question

Gram-Schmidt

If I have an orthonormal basis, how do I verify that they are indeed orthonormal?

I have Q, R and A

is it enough to times Q` by Q to give me I? or A=QR?

Edit:

Let's say I have a Matrix M and I know it is orthonormal basis but I want to verify it

Is $M^{\mathrm{T}}M=\mathrm{I}$ a proof that M are orthonormal basis?

any help?

Answer 1

Please, be more specific when posting the question. What exactly are your matrices $Q$, $R$ and $A$ that you talk about?

One way of checking would be to write down your basis as columns of a matrix, let's call it $M$. Then you simply have to verify that $M^{\mathrm{T}}M=\mathrm{id}$ holds. This would mean that the columns form an ONB.

Answer 2

Check the definition of orthonomality.

A set of vectors $v_1,v_2,v_3,...$ in a given vector space are orthonormal iff $$\forall i,j \left<v_i,v_j \right>=\delta_{ij}$$ i.e. $v_i^Tv_j=1$ if $i=j$ else $0$.

Further, to show that they form orthonormals basis, you have to show that they indeed span your vector space.

Your notation is sloppy. I would not take $A=Q \times R$. Consider $A=Q=R=\mathbf{0}$ Though they satisfy your condition, they are not orthonormal as they don't satisfy standard condition.