Editing Eigenvalue algorithm (section)

===Factorable polynomial equations===

If {{math|''p''}} is any polynomial and {{math|1=''p''(''A'') = 0,}} then the eigenvalues of {{math|''A''}} also satisfy the same equation. If {{math|''p''}} happens to have a known factorization, then the eigenvalues of {{math|''A''}} lie among its roots.

For example, a [[Projection (linear algebra)|projection]] is a square matrix {{math|''P''}} satisfying {{math|1=''P''<sup>2</sup> = ''P''}}. The roots of the corresponding scalar polynomial equation, {{math|1=''λ''<sup>2</sup> = ''λ''}}, are 0 and 1. Thus any projection has 0 and 1 for its eigenvalues. The multiplicity of 0 as an eigenvalue is the [[Kernel (linear algebra)#Representation as matrix multiplication|nullity]] of {{math|''P''}}, while the multiplicity of 1 is the rank of {{math|''P''}}.

Another example is a matrix {{math|''A''}} that satisfies {{math|1=''A''<sup>2</sup> = ''α''<sup>2</sup>''I''}} for some scalar {{math|''α''}}. The eigenvalues must be {{math|±''α''}}. The projection operators
:<math>P_+=\frac{1}{2}\left(I+\frac{A}{\alpha}\right)</math>

:<math>P_-=\frac{1}{2}\left(I-\frac{A}{\alpha}\right)</math>
satisfy
:<math>AP_+=\alpha P_+ \quad AP_-=-\alpha P_-</math>
and
:<math>P_+P_+=P_+ \quad P_-P_-=P_- \quad P_+P_-=P_-P_+=0.</math>

The [[column space]]s of {{math|''P''<sub>+</sub>}} and {{math|''P''<sub>−</sub>}} are the eigenspaces of {{math|''A''}} corresponding to {{math|+''α''}} and {{math|−''α''}}, respectively.