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Chen
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Given a matrix A how can I found its minimal polynomial? I know how to find its characteristic polynomial, but how do I reduce it to minimal?
Thanks,
Chen
Thanks,
Chen
If A is a matrix and for every polynomial q such that q(A)=0 p|q for some monic polynomial p, then p is the minimal of A.Chen said:Given a matrix A how can I found its minimal polynomial? I know how to find its characteristic polynomial, but how do I reduce it to minimal?
Thanks,
Chen
A matrix minimal polynomial is a polynomial of least degree that has the given matrix as a root. It is the unique monic polynomial of least degree that annihilates the given matrix.
The matrix minimal polynomial can be calculated by finding the characteristic polynomial of the given matrix, and then factoring out all the repeated roots. The resulting polynomial will be the minimal polynomial.
The matrix minimal polynomial is important because it provides a way to understand the behavior and properties of a matrix. It can also be used to determine the eigenvalues and eigenvectors of a matrix, which are important in many applications.
No, a matrix can only have one minimal polynomial. This is because the minimal polynomial is unique and is defined as the polynomial of least degree that has the given matrix as a root.
The minimal polynomial and the characteristic polynomial are related in that the minimal polynomial always divides the characteristic polynomial. This means that the characteristic polynomial can be factored into a product of minimal polynomials.