I have two projectors $P, Q$ onto the same subspace $L \subset \mathbb{R^n}$ which are defined by their matrices $n \times n$. And it's said that there are such projectors that $\ker P \neq \ker Q$

But I can't imagine such situation and think up such projectors. Could you please help me?

Thank you!


You are probably thinking about orthogonal projections. Take, in the plane, two projections onto the line $y=x$: one is the orthogonal projection, whose kernel, is of course the line $y=-x$ perpendicular to $y=x$, and the other projection is a skewed projection, taking $(x,y)$ to $(x,x)$. The kernel is the $y$-axis.

It is true however that an orthogonal projection is determined by its image, because then the kernel must be the orthogonal complement.

  • $\begingroup$ thanks a lot ! yes, I thought about orthogonal projection... $\endgroup$ – D F Feb 6 '18 at 14:41

Take $n=2$ and the projections given by the matrices $\left(\matrix{1 & 0\\ 0 &0}\right)$ and $\left(\matrix{1 & 1\\ 0 &0}\right)$.

The point is that not all projections are orthogonal projections.

  • $\begingroup$ You're welcome! $\endgroup$ – Arnaud Mortier Feb 6 '18 at 14:35

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