# Proving AB & BA Have Same Characteristic Polynomial - Simple Ways

• td21
In summary, the characteristic polynomial of a matrix AB can be found by taking the determinant of AB-xI, and this is equal to the determinant of BA-xI. This holds true even if A is not invertible, as long as the equation det(A-lambda I)=0 has finitely many solutions.

If you're happy that Det(AB)=Det(BA), then with I the identity matrix,
the characteristic polynomial is

p(x) = Det(AB-xI) = Det(IAB-xI2) = Det(B-1BAB-xB-1BI)
= Det(BABB-1-xBIB-1) = Det(BA-xI)

If A is invertible, than it is evident that AB and BA are similar matrices, therefore they have the same characteristic polynomial. Otherwise, we notice that the equation in lambda, det(A-lambda I)=0 has finitely many solutions. We can take epsilon such that, for all lambda 0<|lambda|<epsilon, A-lambda I is invertible. Therefore, (A-lambda I)B and B(A-lambda I) must have the same characteristic equation. Also, det(xI-(A-lambda I)B)=det(xI-B(A-lambda I)). For fixed x, each side of this equation is a polynomial in lambda, hence it is continuous. We can take lambda->0 and we are done.

## What is the definition of a characteristic polynomial?

A characteristic polynomial is a polynomial that is used to find the eigenvalues of a square matrix. It is calculated by taking the determinant of the matrix and setting it equal to 0.

## Why is it important to prove that AB and BA have the same characteristic polynomial?

Proving that AB and BA have the same characteristic polynomial is important because it shows that the two matrices have the same eigenvalues. This is useful in many applications, such as in physics and engineering.

## What are some simple ways to prove that AB and BA have the same characteristic polynomial?

One simple way is to use the fact that the characteristic polynomial is invariant under similarity transformations. Another way is to use the Cayley-Hamilton theorem, which states that a matrix satisfies its own characteristic polynomial.

## Can AB and BA have different characteristic polynomials?

No, AB and BA cannot have different characteristic polynomials. This is because the characteristic polynomial is uniquely determined by the matrix and is not affected by changing the order of multiplication.

## What are some real-world applications of proving that AB and BA have the same characteristic polynomial?

Real-world applications include solving systems of linear equations, finding eigenvalues and eigenvectors, and understanding the behavior of dynamic systems in physics and engineering. It is also useful in data analysis and machine learning algorithms.

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