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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
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7.5 Normal Matrices
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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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