How do you find the minimal polynomial of a number?
If F = Q, E = R, α = √2, then the minimal polynomial for α is a(x) = x2 − 2. The base field F is important as it determines the possibilities for the coefficients of a(x). For instance, if we take F = R, then the minimal polynomial for α = √2 is a(x) = x − √2.
What is meant by minimal polynomial?
In linear algebra the minimal polynomial of an algebraic object is the monic polynomial of least degree which that object satisfies. Examples include the minimal polynomial of a square matrix, an endomorphism of a vector space or an algebraic number. The general setting is an algebra A over a field F.
How do you find the minimal polynomial of a 4×4 matrix?
If d = 1, then mipoly = x + b, therefore, M + b*Id = 0 but this is not true. If d = 2, then mipoly = xˆ2 + ax + b, therefore, Mˆ2 + aM + bId = 0 after doing some calculations, I came to the conclusion that the system has no non-zero solutions since the rank of the matrix is 3.
How do you find the Cyclotomic polynomial?
with largest prime divisor p = pk, by repeated polynomial division, as detailed in Algorithm 1.
- Algorithm 1: Calculating Φn(z) by repeated division.
- Input: n = p1p2 ··· pk, a product of k distinct primes.
- Output: Φn(z), the nth cyclotomic polynomial.
- for j = 1 to k do.
How do you write a minimal polynomial?
For example, if A is a multiple aIn of the identity matrix, then its minimal polynomial is X − a since the kernel of aIn − A = 0 is already the entire space; on the other hand its characteristic polynomial is (X − a)n (the only eigenvalue is a, and the degree of the characteristic polynomial is always equal to the …
How do you find the characteristic and minimal polynomial?
The characteristic polynomial of A is the product of all the elementary divisors. Hence, the sum of the degrees of the minimal polynomials equals the size of A. The minimal polynomial of A is the least common multiple of all the elementary divisors.
How do you find a minimal polynomial example?
- Hence x(x2−4) divides the minimal polynomial,
- Hence all these implies that the minimal polynomial is either x(x2−4) or x2(x2−4).
- Now by putting the matrix in the equation x(x2−4) if it comes 0 then x(x2−4) is the minimal polynomial else x2(x2−4) is the minimal polynomial.
What is the degree of minimal polynomial?
The minimal polynomial is always well-defined and we have deg µA(X) ≤ n2. If we now replace A in this equation by the undeterminate X, we obtain a monic polynomial p(X) satisfying p(A) = 0 and the degree d of p is minimal by construction, hence p(X) = µA(X) by definition.