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More Exercises on Matrix Operations and Applications
1. Using Cramer’ s rule , find x_{3} in the solution
to the following system of equations:
2. Multiply out the quadratic form in
variables x _{1} and x_{2} defined by
.
3. For the matrix
(a) Compute all the principal minors of A. What do these tell you about whether
A
is positive or negative definite or semidefinite?
(b) Find the eigenvalues by finding all roots of the
characteristic equation A−rI =
0. What do these tell you about whether A is positive or negative definite or
semidefinite?
(c) For the principal (largest) eigenvalue r_{1}
of A, find the eigenvector by finding a
solution to the system of equations (A − r_{1}I)h = 0. (Normalize this
eigenvector
to have length 1.)
4. Consider a Markov process with two states in which:
• From state 1, an individual moves to state 2 with
probability 3/4 and remains in
state 1 with probability 1/4;
• From state 2, an individual moves to state 1 with
probability 1/2 and remains in
state 2 with probability 1/2.
For this process, do exercises (a) through (c) below:
(a) What is the transition matrix of this process?.
(b) In a large population whose individuals obey these
transition probabilities in
periods t = 1, 2, . . ., suppose that at t = 1 half the population are in state
1 and
half are in state 2. What is the distribution across states at t = 2?
(c) Use eigenvectors to find a stable state of this Markov
process.
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