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Projection (linear algebra)
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== Canonical forms == Any projection <math>P=P^2</math> on a vector space of dimension <math>d</math> over a [[field (mathematics)|field]] is a [[diagonalizable matrix]], since its [[minimal polynomial (linear algebra)|minimal polynomial]] divides <math>x^2-x</math>, which splits into distinct linear factors. Thus there exists a basis in which <math>P</math> has the form :<math>P = I_r\oplus 0_{d-r}</math> where <math>r</math> is the [[rank of a linear transformation|rank]] of <math>P</math>. Here <math>I_r</math> is the [[identity matrix]] of size <math>r</math>, <math>0_{d-r}</math> is the [[zero matrix]] of size <math>d-r</math>, and <math>\oplus</math> is the [[Matrix addition#Direct sum|direct sum]] operator. 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<ref>{{ cite journal|last=Doković|first= D. Ž. |title=Unitary similarity of projectors|journal=[[Aequationes Mathematicae]]| volume =42| issue= 1| pages= 220–224|date=August 1991|doi=10.1007/BF01818492|s2cid= 122704926 }}</ref> :<math>P = \begin{bmatrix}1&\sigma_1 \\ 0&0\end{bmatrix} \oplus \cdots \oplus \begin{bmatrix}1&\sigma_k \\ 0&0\end{bmatrix} \oplus I_m \oplus 0_s.</math> where <math>\sigma_1 \geq \sigma_2\geq \dots \geq \sigma_k > 0</math>. The [[integer]]s <math>k,s,m</math> and the real numbers <math>\sigma_i</math> are uniquely determined. <math>2k+s+m=d</math>. The factor <math>I_m \oplus 0_s</math> corresponds to the maximal invariant subspace on which <math>P</math> acts as an ''orthogonal'' projection (so that ''P'' itself is orthogonal if and only if <math>k=0</math>) and the <math>\sigma_i</math>-blocks correspond to the ''oblique'' components.
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