Homework 5

1

By hand, find eigensystems for each of these matrices. Then specify the algebraic and geometric multiplicity of each eigenvalue.

1. $\left[\begin{array}{rr}7& -10\\ 5& -8\end{array}\right]$

2. $\left[\begin{array}{rrr}-1& 0& 0\\ 1& -1& 0\\ 0& 1& -1\end{array}\right]$

3. $\left[\begin{array}{lll}2& 1& 1\\ 0& 3& 1\\ 0& 0& 2\end{array}\right]$

4. $\left[\begin{array}{ll}0& 2\\ 2& 0\end{array}\right]$

5. $\left[\begin{array}{rr}0& -2\\ 2& 0\end{array}\right]$

2

Compute the eigensystems of these matrices, and identify any defective matrices. (You can do this by hand or on the computer.)

1. $\left[\begin{array}{lll}2& 0& 1\\ 0& 0& 0\\ 1& 0& 2\end{array}\right]$

2. $\left[\begin{array}{lll}2& 0& 0\\ 0& 3& 1\\ 0& 6& 2\end{array}\right]$

3. $\left[\begin{array}{rr}1+\mathrm{i}& 3\\ 0& \mathrm{i}\end{array}\right]$

4. $\left[\begin{array}{rrr}1& -2& 1\\ -2& 4& -2\\ 0& 0& 1\end{array}\right]$

5. $\left[\begin{array}{rrrr}2& 1& -1& -2\\ 0& 1& -1& -2\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right]$

3

By hand, find a matrix $P$ such that $P^{-1}AP$ is diagonal for each of the following matrices. Then use $P$ to find a formula for $A^k,k$ a positive integer.

1. $\left[\begin{array}{lll}2& 0& 1\\ 0& 1& 0\\ 0& 0& 3\end{array}\right]$

2. $\left[\begin{array}{lll}1& 2& 2\\ 0& 0& 0\\ 0& 2& 2\end{array}\right]$

3. $\left[\begin{array}{ll}1& 2\\ 3& 2\end{array}\right]$

4. $\left[\begin{array}{ll}0& 2\\ 2& 0\end{array}\right]$

5. $\left[\begin{array}{rrrr}2& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 3& 1\\ 0& 0& 0& 1\end{array}\right]$

4

Compute $\sin \left(\frac{\pi }{6}A\right)$ and $\cos \left(\frac{\pi }{6}A\right)$, where $A=\left[\begin{array}{rr}2& 4\\ 0& -3\end{array}\right]$.

5

By hand, find the spectral radius and dominant eigenvalue, if any, for each of these matrices:

1. $\left[\begin{array}{rr}-7& -6\\ 9& 8\end{array}\right]$

2. $\frac{1}{3}\left[\begin{array}{ll}1& 3\\ 2& 0\end{array}\right]$

3. $\left[\begin{array}{lll}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]$

4. $\frac{1}{2}\left[\begin{array}{lll}1& 0& 1\\ 1& 0& 0\\ 0& 2& 1\end{array}\right]$

5. $\left[\begin{array}{rr}1& 1\\ -1& -1\end{array}\right]$

6

If the matrices of the previous exercise are transition matrices, for which do all ${\mathbf{x}}^{(k)}$ remain bounded as $k\to \mathrm{\infty }$ ? Are any of these matrices stable?

7

The three-stage insect model of Example 2.21 yields a transition matrix

$$A=\left[\begin{array}{ccc}0.2& 0& 0.25\\ 0.6& 0.3& 0\\ 0& 0.6& 0.8\end{array}\right]$$

Use a technology tool to calculate the eigenvalues of this matrix. Deduce that $A$ is diagonalizable and determine the approximate growth rate from one state to the next (after much time has passed), given a random initial vector.

8

The financial model of Example 2.27 gives rise to a discrete dynamical system $x^{(k+1)}=A{x}^{(k)}$, where the transition matrix is

$$A=\left[\begin{array}{rrr}1& 0.06& 0.12\\ 1& 0& 0\\ 0& 1& 0\end{array}\right]$$

Use a technology tool to calculate the eigenvalues of this matrix. Deduce that $A$ is diagonalizable and determine the approximate growth rate from one state to the next, given a random initial vector.

Find a starting vector $x^{(0)}$ such that growth rate for the first few iterations is quite different from this approximate growth rate, and demonstrate this using a technology tool.

(If you do this using Sympy, and you set up the matrix A using rational numbers, you can get exact results and find a starting vector which will never reach your approximate growth rage. You can see how this result changes if you use floating point numbers at some point in the calculation, e.g. x = N(x).)

from sympy import Matrix, Rational

# Set up the matrix A in symbolic form
A = Matrix([[1, Rational(6, 100), Rational(12, 100)], [1, 0, 0], [0, 1, 0]])

9

On this problem,you may use a computer to find eigensystems etc, but don’t solve it using simulations and trial and error.

A species of bird can be divided into three age groups: age less than 2 years for group 1, age between 2 and 4 years for group 2, and age between 4 and 6 years for the third group. Assume that these birds have at most a 6-year life span. It is estimated that the survival rates for birds in groups 1 and 2 are $50\mathrm{\%}$ and 75%, respectively. Also, birds in groups 1, 2, and 3 produce 0,1 , and 3 offspring on average in any biennium (period of 2 years).

1. Model this bird population as a discrete dynamical system and analyze the long-term change in the population – give a percentage by which the population of group 1 changes from year to year.

2. Now suppose that the survival rates for group 1 and 2 are not known (so are no longer assumed to be 50% and 75%), but are assumed to be equal to one another. What value for this survival rate would make the population stable in the long run?