Projection (linear algebra)
Spectrum

In infinite-dimensional vector spaces, the spectrum of a projection is contained in { 0, 1 } {\displaystyle \{0,1\}} as (λ I − P) − 1 = 1 λ I + 1 λ (λ − 1) P. {\displaystyle (\lambda I-P)^{-1}={\frac {1}{\lambda }}I+{\frac {1}{\lambda (\lambda -1)}}P.} Only 0 or 1 can be an eigenvalue of a projection. This implies that an orthogonal projection P {\displaystyle P} is always a positive semi-definite matrix. In general, the corresponding eigenspaces are (respectively) the kernel and range of the projection. Decomposition of a vector space into direct sums is not unique. Therefore, given a subspace V {\displaystyle V}, there may be many projections whose range (or kernel) is V {\displaystyle V}.

If a projection is nontrivial it has minimal polynomial x 2 − x = x (x − 1) {\displaystyle x^{2}-x=x(x-1)}, which factors into distinct linear factors, and thus P {\displaystyle P} is diagonalizable.
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