In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1.
Theorem(Eigenvalues are roots of the characteristic polynomial) Let A be an n × n matrix, and let f ( λ )= det ( A − λ I n ) be its characteristic polynomial. Then a number λ 0 is an eigenvalue of A if and only if f ( λ 0 )= 0.
The characteristic polynomial pA(t) of a n×n matrix is monic (its leading coefficient is 1) and its degree is n.
To put a matrix in rational canonical form, you find the invariant factors of the matrix, then take the matrix of block matrices consisting of companion matrices for the invariant factors. For A, the invariant factors are x−2 which has a companion matrix [2] and (x−2)(x−3)=x2−5x+6 which has a companion matrix [0−615].
If k is a splitting field of the characteristic polynomial of A, then the elementary divisors have the form (x−λ)m. Their number is then the same as the number of Jordan cells in the Jordan form of A, and the elementary divisor (x−λ)m corresponds to a Jordan cell Jm(λ) of order m( see Jordan matrix).
monic polynomial ~ A Maths Dictionary for Kids Quick Reference by Jenny Eather. a polynomial whose leading coefficient is 1, that is, the coefficient of the first term equals 1.
Companion FormA companion form contains the coefficients of a corresponding characteristic polynomial along one of its far rows or columns.
Definition: For a graph , the Maximum Degree of denoted by , is the degree of the vertex with the greatest number of edges incident to it. The Minimum Degree of denoted by , is the degree of the vertex with the least number of edges incident to it.
Use long division or other arguments to show that none of these is actually a factor. If a polynomial with degree 2 or higher is irreducible in , then it has no roots in . If a polynomial with degree 2 or 3 has no roots in , then it is irreducible in .
A primitive polynomial is a polynomial that generates all elements of an extension field from a base field. Primitive polynomials are also irreducible polynomials. For any prime or prime power and any positive integer , there exists a primitive polynomial of degree over GF( ).
In algebra, the content of a polynomial with integer coefficients (or, more generally, with coefficients in a unique factorization domain) is the greatest common divisor of its coefficients. The primitive part of such a polynomial is the quotient of the polynomial by its content.
: a polynomial expressible as the product of two or more polynomials of lower degree.
2 Answers. Let g be a primitive root of 83. Then all the primitive roots are gk where k is relatively prime to 82, so all gk with odd k from k=1 to k=81, with the exception of k=41.
In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF(q) is called a primitive element if it is a primitive (q − 1)th root of unity in GF(q); this means that each non-zero element of GF(q) can be written as αi for some integer i.
