minimal polynomial (endomorphism)
. Firstly, is a vector space of dimension . Therefore the vectors, , are linearly dependant. So there are coefficients, not all zero such that . We conclude that a non-trivial zero polynomial for exists. We take to be a zero polynomial for of minimal degree with leading coefficient one.
: If is a zero polynomial for then .
The minimal polynomial has a number of interesting properties:
The roots are exactly the eigenvalues of the endomorphism
If the minimal polynomial of splits into linear factors then is upper-triangular with respect to some basis
The minimal polynomial of splits into distinct linear factors (i.e. no repeated roots) if and only if is diagonal with respect to some basis.
The minimal polynomial is intimately related to the characteristic polynomial for . For let be the characteristc polynomial. Since , we have by the above lemma that . It is also a fact that the eigenvalues of are exactly the roots of . So when split into linear factors the only difference between and is the algebraic multiplicity of the roots.
In general they may not be the same - for example any diagonal matrix with repeated eigenvalues.
|Title||minimal polynomial (endomorphism)|
|Date of creation||2013-03-22 13:10:14|
|Last modified on||2013-03-22 13:10:14|
|Last modified by||mathcam (2727)|