Canonical Forms
Any projection P = P2 on a vector space of dimension d over a field is a diagonalizable matrix, since its minimal polynomial is x2 − x, which splits into distinct linear factors. Thus there exists a basis in which P has the form
where r is the rank of P. Here Ir is the identity matrix of size r, and 0d−r is the zero matrix of size d − r. If the vector space is complex and equipped with an inner product, then there is an orthonormal basis in which the matrix of P is
- .
where σ1 ≥ σ2 ≥ ... ≥ σk > 0. The integers k, s, m and the real numbers are uniquely determined. Note that 2k + s + m = d. The factor Im ⊕ 0s corresponds to the maximal invariant subspace on which P acts as an orthogonal projection (so that P itself is orthogonal if and only if k = 0) and the σi-blocks correspond to the oblique components.
Read more about this topic: Projection (linear Algebra)
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