Let $V=M^{R}_{4X4}$ and A a matrix in V. How can I determine weather exists a sub-space of V such as A's orthogonal projection on it has a norm of x?
for example, if $A=\begin{bmatrix}1&-1&1&-1 \\ -1&1&-1&1 \\ 1&-1&1&-1 \\ -1&1&-1&1\end{bmatrix}$ and x=5.
I tried to use the fact that A's norm is 4, and therefore it's projection's norm can't be larger. On a second thought that is obviously incorrect, just like $(1,0)$'s projection on $Sp\{(1,1)\}$ is $(1,1)$ which has a larger norm than $(1,0)$. So I'm stuck, and would love tips on how to approach this (in the specific case or in the general case).
Edit- the orthogonal projection of $(1,0)$ on $Sp\{(1,1)\}$ is not $(1,1)$, it is $(\frac{1}{\sqrt2},\frac{1}{\sqrt2})$, which has a norm of 1 (Thanks Michael Lee). So again I think that the norm of the orthogonal projection is always smaller than the norm of the original vector, but I don't know if it is true and if it is, how to prove it.
Thanks in advance.