A characteristic polynomial is a polynomial equation that is used to find the eigenvalues of a matrix. It is formed by taking the determinant of the matrix and subtracting the identity matrix multiplied by a variable, usually denoted as lambda.
Linear factors are polynomial expressions of degree one, such as (x+2) or (3x-5). When a polynomial splits into linear factors, it means that the polynomial can be written as a product of these linear factors.
A characteristic polynomial splits into linear factors when it has distinct eigenvalues. This means that each eigenvalue has a corresponding eigenvector, and the polynomial can be written as a product of (lambda - eigenvalue) terms.
When a characteristic polynomial splits into linear factors, it allows for easier computation of the eigenvalues of a matrix. It also allows for the diagonalization of a matrix, which is useful in solving systems of linear equations and other applications in mathematics and science.
Yes, there are cases where a characteristic polynomial may not split into linear factors. This can happen when the matrix has repeated eigenvalues or when the eigenvalues are complex numbers. In these cases, the polynomial may still be factored but into irreducible quadratic or higher degree terms.