Algebraic multiplicity
Algebraic multiplicity Let λ i be an eigenvalue of an n by n matrix A . The algebraic multiplicity μ A ( λ i ) of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer k such that ( λ − λ i ) k divides evenly that polynomial. [9] [25] [26] Suppose a matrix A has dimension n and d ≤ n distinct eigenvalues. Whereas Equation ( 4 ) factors the characteristic polynomial of A into the product of n linear terms with some terms potentially repeating, the characteristic polynomial can instead be written as the product of d terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity, | � − � � | = ( � 1 − � ) � � ( � 1 ) ( � 2 − � ) � � ( � 2 ) ⋯ ( � � − � ) � � ( � � ) . If d = n then the ri...