Can you guys please help me with this question: Determine the eigenvalues of the linear transformation. $P: \mathbb R^3 \to \mathbb R^3$ given by projection onto the plane $3x-y-2z=0$.
2026-04-01 17:44:52.1775065492
Linear Transformation eigenvalues: from $\mathbb R^3$ to a plane
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Linear transformation is a projection $P$ then eigenvalues can be only $0$ and $1$ ($P^2=P$).
Because it is projection into two-dimensional subspace (the plane) so there is one zero eigenvalue and two eigenvalues $=1$.
As you see what is important in this case is only that it is a projection onto a plane, specific equation of the plane is not so important for eigenvalues - it is however important for eigenvectors.
Notice that for eigenvalue $= 1$ and its eigenvector $v \ \ $ we have equation $Pv=v$ what means that for this eigenvalue the eigenvector is unchanged by the projection, this is a case for vectors lying in the plane.