Editing Characteristic polynomial (section)

==Motivation==
In [[linear algebra]], [[eigenvalues and eigenvectors]] play a fundamental role, since, given a [[linear transformation]], an eigenvector is a vector whose direction is not changed by the  transformation, and the corresponding eigenvalue is the measure of the resulting change of magnitude of the vector.

More precisely, suppose the transformation is represented by a square matrix <math>A.</math> Then an eigenvector <math>\mathbf{v}</math> and the corresponding eigenvalue <math>\lambda</math> must satisfy the equation
<math display=block>A \mathbf{v} = \lambda \mathbf{v},</math>
or, equivalently (since  <math>\lambda \mathbf{v} = \lambda I \mathbf{v}</math>),
<math display=block>(\lambda I - A) \mathbf{v} =\mathbf 0</math>
where <math>I</math>  is the [[identity matrix]], and <math>\mathbf{v}\ne \mathbf{0}</math>
(although the zero vector satisfies this equation for every <math>\lambda,</math> it is not considered an eigenvector). 

It follows that the matrix <math>(\lambda I - A)</math> must be [[singular matrix|singular]], and its determinant
<math display=block>\det(\lambda I - A) = 0</math>
must be zero.

In other words, the eigenvalues of {{mvar|A}} are the [[zero of a function|roots]] of 
<math display=block>\det(xI - A),</math>
which is a [[monic polynomial]] in {{mvar|x}} of degree {{mvar|n}} if {{mvar|A}} is a {{math|''n''×''n''}} matrix. This polynomial is the ''characteristic polynomial'' of {{mvar|A}}.