Representation Theory
The characteristic polynomial is a polynomial that is derived from a square matrix, encapsulating essential information about the linear transformation represented by that matrix. Specifically, it is obtained from the determinant of the matrix subtracted by a variable multiplied by the identity matrix, expressed as $\text{det}(A - \lambda I) = 0$, where $\lambda$ represents the eigenvalues of the matrix. The roots of this polynomial give the eigenvalues, linking deeply with characters in representation theory.
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