# 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. 12/08/2019