Can two matrices have the same characteristic polynomial?

Two similar matrices have the same characteristic polynomial. The converse however is not true in general: two matrices with the same characteristic polynomial need not be similar.

How many matrices have the same characteristic polynomial?

two similar matrices
by what we have previously done. In other words,any two similar matrices have the same characteristic polynomial.

When minimal polynomial is equal to characteristic polynomial?

Let χ(t) be the characteristic polynomial of T, and assume that the factorization of χ(t) into irreducibles over F is χ(t)=ϕ1(t)a1⋯ϕk(t)ak. Then the minimal polynomial of T equals the characteristic polynomial of T if and only if dim(ker(ϕi(T))=deg(ϕi(t)) for i=1,…,k.

How do you know if two matrices are similar?

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

Are two matrices similar?

If two matrices are similar, they have the same eigenvalues and the same number of independent eigenvectors (but probably not the same eigenvectors). If two matrices have the same n distinct eigenvalues, they’ll be similar to the same diagonal matrix.

Do similar matrices have the same determinant?

Similar matrices have the same rank, the same determinant, the same characteristic polynomial, and the same eigenvalues.

What is the difference between minimal polynomial and characteristic polynomial?

That is to say, if A has minimal polynomial m(t) then m(A)=0, and if p(t) is a nonzero polynomial with p(A)=0 then m(t) divides p(t). The characteristic polynomial, on the other hand, is defined algebraically. If A is an n×n matrix then its characteristic polynomial χ(t) must have degree n.

Is the 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.

How do you prove that similar matrices have the same characteristic polynomial?

– Mathematics Stack Exchange Elegant proofs that similar matrices have the same characteristic polynomial? It’s a simple exercise to show that two similar matrices has the same eigenvalues and eigenvectors (my favorite way is noting that they represent the same linear transformation in different bases).

How to prove that $AB$ and $BA$ have the same characteristic polynomial?

The same goes if $B$ is invertible. In general, from the above observation, it is not too difficult to show that $AB$, and $BA$ have the same characteristic polynomial, the type of proof could depends on the field considered for the coefficient of your matrices though.

Do a B and B A have the same minimal polynomial?

In general A B and B A do not have the same minimal polynomial. I’ll let you search a bit for a counter example. Show activity on this post. therefore easily conclude if m = n then A B and B A have the same characteristic polynomials. C = [ x I m A B I n], D = [ I m 0 − B x I n].