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.

How do you find the characteristic polynomial of a 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.

Is a minimal polynomial unique?

In field theory, a branch of mathematics, the minimal polynomial of an element α of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the field, such that α is a root of the polynomial. If the minimal polynomial of α exists, it is unique.

Are matrices with the same eigenvalues similar?

Two similar matrices have the same eigenvalues, even though they will usually have different eigenvectors. Said more precisely, if B = Ai’AJ. I and x is an eigenvector of A, then M’x is an eigenvector of B = M’AM. Also, if two matrices have the same distinct eigen values then they are similar.

What do similar matrices have in common?

If two matrices are similar, they have the same eigenvalues and the same number of independent eigenvectors (but probably not the same eigenvectors).

Can two similar matrices have different minimal polynomials?

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 …

How do you find the minimal polynomial of a matrix?

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.

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

Starts here6:06Example of Minimal Polynomial – YouTubeYouTube

What is the minimal polynomial of Nilpotent matrix?

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

How do you find the similarity of a matrix?

Definition (Similar Matrices) Suppose A and B are two square matrices of size n . Then A and B are similar if there exists a nonsingular matrix of size n , S , such that A=S−1BS A = S − 1 B S .