54 alternate max min and min max formulas are given

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Unformatted text preview: 200 of species II. (a) Determine the number of each species at all future times. (b) Determine which species is destined to become extinct, and compute the time to extinction. Y P O C 7.4.4. Cooperating Species. Consider two species that survive in a symbiotic relationship in the sense that the population of each species decreases at a rate equal to its existing number but increases at a rate equal to the existing number in the other population. (a) If there are initially 200 of species I and 400 of species II, determine the number of each species at all future times. (b) Discuss the long-run behavior of each species. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu Buy from AMAZON.com 7.5 Normal Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 7.5 NORMAL MATRICES 547 A matrix A is diagonalizable if and only if A possesses a complete independent set of eigenvectors, and if such a complete set is used for columns of P, then P−1 AP = D is diagonal (p. 507). But even when A possesses a complete independent set of eigenvectors, there’s no guarantee that a complete orthonormal set of eigenvectors can be found. In other words, there’s no assurance that P can be taken to be unitary (or orthogonal). And the Gram–Schmidt procedure (p. 309) doesn’t help—Gram–Schmidt can turn a basis of eigenvectors into an orthonormal basis but not into an orthonormal basis of eigenvectors. So when (or how) are complete orthonormal sets of eigenvectors produced? In other words, when is A unitarily similar to a diagonal matrix? D E T H Unitary Diagonalization A ∈ C n×n is unitarily similar to a diagonal matrix (i.e., A has a complete orthonormal set of eigenvectors) if and only if A∗ A = AA∗ , in which case A is said to be a normal matrix. • Whenever U∗ AU = D with U unitary and D diagonal, the columns of U must be a complete orthonormal set of eigenvectors for A, an...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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