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td21
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For two matrices A and B, when p(A)=0 iff p(B)=0 for any polynomial, what will happen? i read that A and B have the same minimal polynomial, why?
td21 said:For two matrices A and B, when p(A)=0 iff p(B)=0 for any polynomial, what will happen? i read that A and B have the same minimal polynomial, why?
When p(A) equals 0, it means that when the polynomial p(x) is evaluated at the value A, the result is 0. In other words, A is a root or solution of the polynomial equation p(x) = 0.
If both p(A) and p(B) equal 0, it means that both A and B are roots of the polynomial equation p(x) = 0. This means that both A and B are solutions to the same polynomial equation.
This property is important because it allows us to compare and relate different roots of a polynomial. If two different values, A and B, both make the polynomial equal to 0, then it means they are both solutions to the same equation and can be used interchangeably in certain situations.
The minimal polynomial is the smallest degree polynomial that has a given root or solution. In other words, it is the polynomial of lowest degree that has the same root as a given polynomial.
The same minimal polynomial matters because it provides a unique representation of a root or solution. In other words, if two different polynomials have the same root, then they must also have the same minimal polynomial. This allows us to make certain conclusions and simplifications when working with polynomial equations.