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12 - Eigen 4 Characteristic Polynomials

12 - Eigen 4 Characteristic Polynomials - 1 Math 1b...

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1 Math 1b Practical — Eigenvalues and eigenvectors, IV: The characteristic polynomial February 16, 2011 We have talked about the geometric multiplicity of eigenvalues. There is another kind of multiplicity. We know the λ is an eigenvalue of A if and only if p A ( λ ) = 0 where p A ( x ) = det( xI A ) is the characteristic polynomial of A . For example, if A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 , then p A ( x ) = x a 11 a 12 a 13 a 21 x a 22 a 23 a 31 a 32 x a 33 = x 3 ( a 11 + a 22 + a 33 ) x 2 + ( a 22 a 33 a 23 a 32 + a 11 a 33 a 13 a 31 + a 11 a 22 a 12 a 21 ) x ( a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 a 31 a 22 a 13 a 32 a 23 a 11 a 33 a 21 a 12 ) . (3) By the Fundamental Theorem of Algebra, we can factor any polynomial into linear factors over the complex numbers, so p A ( x ) = ( x λ 1 ) m 1 ( x λ 2 ) m 2 · · · ( x λ ) m (4) where the λ i ’s are distinct complex numbers (the distinct eigenvalues of A ). The expo- nent m i is called the algebraic multiplicity of λ i as an eigenvalue of A . The algebraic multiplicities obviously sum to n . Theorem 5. Let λ be an eigenvalue of A . Then the geometric multiplicity of λ cannot exceed the algebraic multiplicity of λ . A matrix is diagonalizable if and only if the geometric multiplicity of every eigenvalue is equal to its algebraic multiplicity.

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