21-eigenvalues

21-eigenvalues - COMPUTING EIGENVALUES Math 21b O.Knill THE...

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Unformatted text preview: COMPUTING EIGENVALUES Math 21b, O.Knill THE TRACE. The trace of a matrix A is the sum of its diagonal elements. EXAMPLES. The trace of A = 1 2 3 3 4 5 6 7 8 is 1 + 4 + 8 = 13. The trace of a skew symmetric matrix A is zero because there are zeros in the diagonal. The trace of I n is n . CHARACTERISTIC POLYNOMIAL. The polynomial f A ( λ ) = det( A- λI n ) is called the characteristic polynomial of A . EXAMPLE. The characteristic polynomial of A above is- x 3 + 13 x 2 + 15 x . The eigenvalues of A are the roots of the characteristic polynomial f A ( λ ). Proof. If λ is an eigenvalue of A with eigenfunction ~v , then A- λ has ~v in the kernel and A- λ is not invertible so that f A ( λ ) = det( A- λ ) = 0. The polynomial has the form f A ( λ ) = (- λ ) n + tr( A )(- λ ) n- 1 + ··· + det( A ) THE 2x2 CASE. The characteristic polynomial of A = a b c d is f A ( λ ) = λ 2- ( a + d ) / 2 λ + ( ad- bc ). The eigenvalues are λ ± = T/ 2 ± p ( T/ 2) 2- D , where T is the trace and D is the determinant. In order that this is real, we must have ( T/ 2) 2 ≥ D . Away from that parabola, there are two different eigenvalues. The map A contracts volume for | D | < 1....
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