$X$ and $Y$ have the same range(column space). Why there is a unitary $U \in {M_m}$ such that $X = YU$?

Question

Let $X,Y \in {M_{n*m}}$ have orthonormal column. Also $X$ and $Y$ have the same range(column space). Why there is a unitary $U \in {M_m}$ such that $X = YU$?

Answer 1

Since the columns of $Y$ and $X$ are orthonormal and they have the same column space we can take the columns of $Y$ as an orthonormal basis for columns of $X$. This way we get a matrix $M$ such that $X=YM$. Now since they are orthonormal we get a unique such matrix. Say $C_i$ and $C_j$ are two orthonormal columns of $X$. Then we can write them as $C_i=\sum a_k D_k$ and $C_j=\sum b_kD_k$ where $D_k$ are columns of $Y$. Now we need to show that $\sum a_kb_k=0$. This is true since $<C_i,C_j>=\sum a_kb_k$ due to orthonormality of $D_k$ .