# c34 - 18.03 Class 34, May 5 Classification of Linear Phase...

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2/4 18.03 Class 34, May 5 Classification of Linear Phase Portraits The moral of today's lecture: Eigenvalues Rule (usually) [1] Recall that the characteristic polynomial of a square matrix A is p_A(lambda) = det(A - lambda I). In the 2x2 case A = [a b ; c d ] this can be rewritten as p_A(lambda) = lambda^2 - (tr A) lambda + (det A) where tr(A) = a + d , det(A) = ad - bc. Its roots are the eigenvalues, so p_A(lambda) = (lambda - lambda_1)(lambda - lambda_2) = lambda^2 - (lambda_1 + lambda_2) lambda + (lambda_1 lambda_2) Comparing coefficients, tr(A) = lambda_1 + lambda_2 , det(A) = lambda_1 lambda_2 so the two numbers tr(A) and det(A) , extracted from the four numbers a, b, c, d, are determined by the eigenvalues. Conversely, they determine the eigenvalues, as the roots: by the quadratic formula, lambda1,2 = tr(A)/2 +- sqrt(tr(A)^2/4 - det(A)). lambda1,2 are not real if det(A) > tr(A)^2/4 are equal if det(A) = tr(A)^2/4 are real and different from each other if det(A) < tr(A)^ The boundary is the "critical parabola," where det(A) = tr(A)^2/4. Notice that if the eigenvalues are not real, their real part is tr (A)/2. If the eigenvalues are real, they have the same sign exactly when their product is positive, and that sign is positive if their sum is also positive.

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det ^ . |<-----purely imaginary . complex roots . . | . . Re < 0
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## This note was uploaded on 01/31/2011 for the course MAT 17A taught by Professor Staff during the Winter '08 term at UC Davis.

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c34 - 18.03 Class 34, May 5 Classification of Linear Phase...

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