If the columns of $U$ are orthonormal, then $UU^t y$ is the orthogonal projection of $y$ onto the column space of $U$

Question

Problem. Determine if it is true or false :

If the columns of a $n\times p$ matrix $U$ are orthonormal, then $UU^t y$ is the orthogonal projection of the vector $y$ onto the column space of $U$.

Hi, can someone tell me the theorem about this? I would like to Google it myself but i dont know how.

So someone who can explain me this or know what terms I have to Google for in Google?

Thank you!

Answer 1

If the columns of a matrix $A$ are independent, then projection onto $\operatorname{Col}(A)$ is given by $$ P_{\operatorname{Col}(A)}=A(A^\top A)^{-1}A^\top $$ Additionally, if the columns of $A$ are orthonormal, then $A^\top A=I$.

So, if the columns of $A$ are orthonormal, then projection onto $\operatorname{Col}(A)$ is given by $$ P_{\operatorname{Col}(A)} = A(A^\top A)^{-1}A^\top = AI^{-1}A^\top = AA^\top $$