Editing Multiset (section)

==Examples==
One of the simplest and most natural examples is the multiset of [[prime factor]]s of a natural number {{mvar|n}}. Here the underlying set of elements is the set of prime factors of {{mvar|n}}. For example, the number [[120 (number)|120]] has the [[prime factorization]]
<math display="block">120 = 2^3 3^1 5^1,</math>
which gives the multiset {{math|{{mset|2, 2, 2, 3, 5}}}}.

A related example is the multiset of solutions of an [[algebraic equation]]. A [[quadratic equation]], for example, has two solutions. However, in some cases they are both the same number.  Thus the multiset of solutions of the equation could be {{math|{{mset|3, 5}}}}, or it could be {{math|{{mset|4, 4}}}}.  In the latter case it has a solution of multiplicity 2. More generally, the [[fundamental theorem of algebra]] asserts that the [[complex number|complex]] solutions of a [[polynomial equation]] of [[degree of a polynomial|degree]] {{mvar|d}} always form a multiset of cardinality {{mvar|d}}.

A special case of the above are the [[eigenvalue]]s of a [[matrix (mathematics)|matrix]], whose multiplicity is usually defined as their multiplicity as [[root of a polynomial|roots]] of the [[characteristic polynomial]]. However two other multiplicities are naturally defined for eigenvalues, their multiplicities as roots of the [[minimal polynomial (linear algebra)|minimal polynomial]], and the [[geometric multiplicity]], which is defined as the [[dimension (vector space)|dimension]] of the [[Kernel (linear algebra)|kernel]] of {{math|''A'' − ''λI''}} (where {{mvar|λ}} is an eigenvalue of the matrix {{mvar|A}}). These three multiplicities define three multisets of eigenvalues, which may be all different: Let {{mvar|A}} be a {{math|''n''&thinsp;×&thinsp;''n''}} matrix in [[Jordan normal form]] that has a single eigenvalue. Its multiplicity is {{mvar|n}}, its multiplicity as a root of the minimal polynomial is the size of the largest Jordan block, and its geometric multiplicity is the number of Jordan blocks.