<s>[INST] Question on uniqueness of projections in linear algebra Suppose A and B are projection matrices and suppose that for some vector x in the column space of B, BAx=x. Can I say that Ax=x?
Since x is in the column space of B, I know that Bx=x it projects itself in its own column space. I guess my question is, is x the only vector that can give the outcome of x when premultiplied by the projection matrix B?
 [/INST] Let $A = \begin{pmatrix}0&1\\0&1\end{pmatrix},$ $B = \begin{pmatrix}0&0\\0&1\end{pmatrix},$ and $x = \begin{pmatrix}0\\1\end{pmatrix}.$ Then $A^2 = A$ and $B^2 = B,$ so $A$ and $B$ are indeed projection matrices. On the other hand, 
$$
\begin{pmatrix}0&0\\0&1\end{pmatrix}\begin{pmatrix}0&1\\0&1\end{pmatrix}\begin{pmatrix}0\\1\end{pmatrix} = \begin{pmatrix}0&0\\0&1\end{pmatrix}\begin{pmatrix}1\\1\end{pmatrix}= \begin{pmatrix}0\\1\end{pmatrix},
$$
so that $BAx = x,$ but $Ax\neq x.$
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