Three holy men (let’s call them Anselm, Benjamin, and Caspar) put little stock in material things; their only earthly possession is a small purse with a bit of gold dust. Each day they get together for the following bizarre bonding ritual: Each of them takes his purse and gives his gold away to the two others, in equal parts. For example, if Anselm has 4 ounces one day, he will give 2 ounces each to Benjamin and Caspar.
(a) If Anselm starts out with 6 ounces, Benjamin with 1 ounce, and Caspar with 2 ounces, find formulas for the amounts a(t), b(t), and c(t) each will have after t distributions.
Hint: The vector , and will be useful.
(b) Who will have the most gold after one year, that is, after 365 distributions?
The solutions are,
This is dynamical system, with,
This matrix’s eigenvectors are,
With the corresponding eigenvalues being .
We also have
So Benjamin will have the most ounces of gold.
Hence final solution is,
25: Consider a positive transition matrix
meaning that a, b, c, and d are positive numbers such that a + c = b + d = 1. (The matrix in Exercise 24 has this form.) Verify that
are eigenvectors of A. What are the associated eigenvalues? Is the absolute value of these eigenvalues more or less than 1?
Sketch a phase portrait.
In all parts of this problem, let V be the linear space of all 2 × 2 matrices for which is an eigenvector.
(a) Find a basis of V and thus determine the dimension of V.
(b) Consider the linear transformation T (A) = A from V to . Find a basis of the image of T and a basis of the kernel of T. Determine the rank of T .
(c) Consider the linear transformation L(A) = A from V to . Find a basis of the image of L and a basis of the kernel of L. Determine the rank of L.
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