When minimal polynomial and characteristic polynomial are same?

The characteristic polynomial of a square matrix whose eigenvalues are all simple is equal to its minimal polynomial: for example, the eigenvalues of the adjacency matrix of an undirected path graph are all simple, and hence its characteristic polynomial is equal to its minimal polynomial.

Are matrices with the same minimal polynomial similar?

Note that if p(A) = 0 for a polynomial p(λ) then p(C−1AC) = C−1p(A)C = 0 for any nonsingular matrix C; hence similar matrices have the same minimal polynomial, and the characteristic and minimal polynomials of a linear transfor- mation T thus can be defined to be the corresponding polynomials of any matrix representing …

Is the characteristic polynomial the 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 does it mean if two matrices have the same characteristic polynomial?

If A and B are n x n matrices such that there is an invertible n x n matrix P with B = P-1 AP, then A and B are called similar. (We will give a geometric interpretation to similar matrices later.) In other words,any two similar matrices have the same characteristic polynomial.

Does minimal polynomial divides annihilating polynomial?

17. If the characteristic polynomial of an operator is of form f(x) = P 1ฃ iฃ m (x-l i)k(i), (k(i) ณ 1, 1 ฃ i ฃ m), the possibilities for the minimal polynomial p(x) are only from amongst the k(1)ด k(2)ด … ด k(m) polynomials p(x) = P 1ฃ iฃ m (x-l i)j(i), (k(i) ณ j(i) ณ 1, 1 ฃ i ฃ m).

Is the minimal polynomial of a matrix irreducible?

Since fields are also integral domains, we have g(α) = 0 or h(α) = 0. This contradicts the minimality of the degree of f. Thus minimal polynomials are irreducible.

Does the minimal polynomial always exist?

The minimal polynomial is often the same as the characteristic polynomial, but not always.

How do you find the minimal polynomial using the characteristic polynomial?

So if you know the characteristic polynomial P, the minimal polynomial must be obtained by taking every distinct factor of P at least once, and at most as many times as it occurs as factor of P. Any polynomial so obtained (in your case there are 4 of them) can be the minimal polynomial.

How do you find the minimal polynomial of a characteristic polynomial?

What is the minimal polynomial of Nilpotent matrix?

If N is m-nilpotent, then its minimal polynomial is mN (x) = xm .

What are the characteristics of minimal polynomials?

The minimal polynomial is equal to the characteristic polynomial. The list of invariant factors has length one. The Rational Canonical Form has a single block. The operator has a matrix similar to a companion matrix.

Are the polynomials m(x) = (x – 1)2 similar?

Those polynomials cannot be X − 1 as neither M nor N is equal to the identity matrix. However we have μ M ( X) = μ N ( X) = ( X − 1) 2 as can be verified following an easy computation. Finally, the matrices M and N are not similar.

Are the matrices M and N similar?

Finally, the matrices M and N are not similar. If that would be the case, the dimensions for M and N of the eigenspaces corresponding to the eigenvalue 1 would be equal. However we have You must be logged in to post a comment.

How is the operator similar to a companion matrix?

The operator has a matrix similar to a companion matrix. There exists a (so-called cyclic) vector whose images by the operator span the whole space. Point 1. and 2. are equivalent because the minimal polynomial is the largest invariant factor and the characteristic polynomial is the product of all invariant factors.