Question

# If A is an orthogonal matrix, find a QR factorization of A

Polynomial factorization
If A is an orthogonal matrix, find a QR factorization of A

2021-05-31

Calculation:
If a matrix A is an orthogonal matrix,
its column are already orthonormal vectors,
so, we don't need to change them using the Gram-Schmidt process to obtain column for Q.
So, we can take $$A=Q$$
Step 2
now, we can find the matrix R from $$A=QR$$ using the fact that if A orthogonal it is invertible.
$$A=QR=AR$$
$$\Rightarrow A^{-1}A=A^{-1}AR=R$$
$$\Rightarrow R=I$$
So, QR factorization for an orthogonal matrix $$A=AI$$.