Properties
The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity
That is, when we substitute numerical values for the variables, we obtain the monic univariate polynomial (with variable λ) whose roots are the values substituted for and whose coefficients are the elementary symmetric polynomials.
The characteristic polynomial of a linear operator is an example of this. The roots are the eigenvalues of the operator. When we substitute these eigenvalues into the elementary symmetric polynomials, we obtain the coefficients of the characteristic polynomial, which are numerical invariants of the operator. This fact is useful in linear algebra and its applications and generalizations, like tensor algebra and disciplines which extensively employ tensor fields, such as differential geometry.
The set of elementary symmetric polynomials in variables generates the ring of symmetric polynomials in variables. More specifically, the ring of symmetric polynomials with integer coefficients equals the integral polynomial ring (See below for a more general statement and proof.) This fact is one of the foundations of invariant theory. For other systems of symmetric polynomials with a similar property see power sum symmetric polynomials and complete homogeneous symmetric polynomials.
Read more about this topic: Elementary Symmetric Polynomial
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)