Editing Characteristic polynomial (section)

==Formal definition==

Consider an <math>n \times n</math> matrix <math>A.</math> The characteristic polynomial of <math>A,</math> denoted by <math>p_A(t),</math> is the polynomial defined by<ref>{{Cite book|title=Advanced linear algebra |author=Steven Roman |url=https://archive.org/details/springer_10.1007-978-1-4757-2178-2 |isbn=3540978372 |edition=2 |year=1992 |publisher=Springer |page=[https://archive.org/details/springer_10.1007-978-1-4757-2178-2/page/n142 137]}}</ref>
<math display=block>p_A(t) = \det (t I - A)</math>
where <math>I</math> denotes the <math>n \times n</math> [[identity matrix]].

Some authors define the characteristic polynomial to be <math>\det(A - t I).</math> That polynomial differs from the one defined here by a sign <math>(-1)^n,</math> so it makes no difference for properties like having as roots the eigenvalues of <math>A</math>; however the definition above always gives a [[monic polynomial]], whereas the alternative definition is monic only when <math>n</math> is even.