How do you find the minimal polynomial of a diagonal matrix?

How do you find the minimal polynomial of a diagonal matrix?

But for a diagonal matrix, the minimal polynomial is just the product of factors x − λ, where λ runs through the distinct diagonal entries. (It is the monic polynomial of smallest degree that has all diagonal entries as roots.)

Is minimal polynomial Diagonalizable?

More generally, if φ satisfies a polynomial equation P(φ) = 0 where P factors into distinct linear factors over F, then it will be diagonalizable: its minimal polynomial is a divisor of P and therefore also factors into distinct linear factors.

How do you find the 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.

Is the minimal polynomial irreducible?

A minimal polynomial is irreducible. Let E/F be a field extension over F as above, α ∈ E, and f ∈ F[x] a minimal polynomial for α. Suppose f = gh, where g, h ∈ F[x] are of lower degree than f. Thus minimal polynomials are irreducible.

What is meant by Monic polynomial?

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.

What is the difference between characteristic polynomial 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.

What is meant by primitive polynomial?

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( ).

Is 1 a Monic polynomial?

Actually, since the constant polynomial 1 is monic, this semigroup is even a monoid.

How do you find a reducible polynomial?

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 .

Is 2x 1 a monic polynomial?

but p(x,y) is not monic as an element in R[x][y], since then the highest degree coefficient (i.e., the y2 coefficient) is 2x − 1. Notably, the product of monic polynomials again is monic.

Is the inverse of Ti an invertible polynomial?

It is interesting to note that for any invertible T ∞ L(V), its inverse Tî is actually a polynomial in T. This fact is essentially shown in the proof of the next theorem. Theorem 7.5Let V be finite-dimensional over F. Then T ∞ L(V) is invert- ible if and only if the constant term in the minimal polynomial for T is not equal to zero.

Which is a nonzero polynomial of degree less than n?

F[x] with the property that m(T) = 0. If mæ is another distinct monic polyno- mial of degree n with the property that mæ(T) = 0, then m – mæ is a nonzero polynomial of degree less than n (since the leading terms cancel) that is satis- fied by T, thus contradicting the definition of n. This proves the existence of a unique monic minimal polynomial.

Which is the matrix representation of a polynomial?

Theorem 7.1Let A be the matrix representation of an operator T ∞ L(V). Then f(A) is the representation of f(T) for any polynomial f ∞F[x]. ProofThis is Exercise 7.1.1. The basic algebraic properties of polynomials in either operators or matrices are given by the following theorem.

How many polynomials are in 7.1 minimal polynomials?

7.1 MINIMAL POLYNOMIALS297 point of view” is that the proof techniques and way of thinking can be quite different.