Can two matrices have the same characteristic polynomial?

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 do you show two matrices not similar?

Having full rank means its columns are linearly independent, and so if x 0 we must have 0. We see right away that if two matrices have different eigenvalues then they are not similar. Also, if two matrices have the same distinct eigen values then they are similar. Suppose A and B have the same distinct eigenvalues.

Can non Similar matrices have the same eigenvalues?

Two matrices may have the same eigenvalues and the same number of eigen vectors, but if their Jordan blocks are different sizes those matrices can not be similar.

How is λ related to the characteristic polynomial?

Factoring the characteristic polynomial If A is an n × n matrix, then the characteristic polynomial f ( λ ) has degree n by the above theorem. When n = 2, one can use the quadratic formula to find the roots of f ( λ ) .

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 of a 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 check if two matrices are similar in Matlab?

A == B will return a matrix the same size as A (and B) telling you which elements of A are equal to the corresponding element of B. isequal(A, B) will return a single true/false telling you if all the elements of A are the same as the ones in B.

Can two different matrices have the same determinant?

Thus, both the matrices have the same determinant value. Hence, we cay say, two different matrices can have the same determinant value.

What is the formula for characteristic polynomial?

We can express the characteristic polynomial as C ( x ) = ( x − λ 1 ) ( x − λ 2 ) ⋯ ( x − λ k ) ⋯ ( x − λ n ) where are the eigenvalues of the matrix .