MATLAB: Is this happening? Symbolic

Question

I calculate the caracteristic polinomial of a matrix in two different ways and the result is not the same, it is opposite.

I have a matrix M:

syms v w  x  y  z lambda
M=[v+w 8 x*3;
    y-z 1 4*x;
    z 3 4];

First I calculate the caracterisc polinomial by:

I=eye(size(M));
Pcaratt=det(M-lambda*I);

Then I do it by:

polinomio=poly(M,lambda);

Finally, if I do:

aux=Pcaratt-polinomio;

the result is not zero. Insted of that, doing:

aux=-Pcaratt-polinomio;

I get zero. Why Pcaratt and polinomio are opposite?

Thank you very much.

Answer 1

The definition of characteristic polynomial seems to vary. For example, Wolfram World defines it the way you did, but Wikipedia would define it as

Pcaratt = det(lambda*I-M);

Then you would get Pcaratt=polinomio. Either way, the equations have the same roots, so the choice of sign is not very important.