21-eigenvalues

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

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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 ( ) = det( A- I 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 ( ). Proof. If is an eigenvalue of A with eigenfunction ~v , then A- has ~v in the kernel and A- is not invertible so that f A ( ) = det( A- ) = 0. The polynomial has the form f A ( ) = (- ) n + tr( A )(- ) n- 1 + + det( A ) THE 2x2 CASE. The characteristic polynomial of A = a b c d is f A ( ) = 2- ( a + d ) / 2 + ( ad- bc ). The eigenvalues are = 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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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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