Többszöriséget

Többszöriséget refers to the concept of multiplicity in mathematics. It is a property associated with roots of polynomials or eigenvalues of matrices. When a root or eigenvalue occurs more than once, it is said to have a multiplicity greater than one.

For instance, consider the polynomial $x^2 - 2x + 1 = 0$. This equation can be factored as $(x-1)(x-1) = 0$, or $(x-1)^2 = 0$. The root $x=1$ appears twice. Therefore, the root $1$ has a multiplicity of $2$. If a root appears only once, its multiplicity is $1$, which is often referred to as a simple root.

In the context of linear algebra, when analyzing the eigenvalues of a matrix, a repeated eigenvalue indicates a geometric or algebraic multiplicity greater than one. The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial. The geometric multiplicity is the dimension of the corresponding eigenspace. These multiplicities are related and provide important information about the structure of the matrix and the linear transformation it represents. Understanding többszöriséget is crucial for solving systems of linear equations, analyzing differential equations, and in various areas of abstract algebra.