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]])Homework 5
1
By hand, find eigensystems for each of these matrices. Then specify the algebraic and geometric multiplicity of each eigenvalue.
\(\left[\begin{array}{rr}7 & -10 \\ 5 & -8\end{array}\right]\)
\(\left[\begin{array}{rrr}-1 & 0 & 0 \\ 1 & -1 & 0 \\ 0 & 1 & -1\end{array}\right]\)
\(\left[\begin{array}{lll}2 & 1 & 1 \\ 0 & 3 & 1 \\ 0 & 0 & 2\end{array}\right]\)
\(\left[\begin{array}{ll}0 & 2 \\ 2 & 0\end{array}\right]\)
\(\left[\begin{array}{rr}0 & -2 \\ 2 & 0\end{array}\right]\)
2
Compute the eigensystems of these matrices, and identify any defective matrices. (You can do this by hand or on the computer.)
\(\left[\begin{array}{lll}2 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 2\end{array}\right]\)
\(\left[\begin{array}{lll}2 & 0 & 0 \\ 0 & 3 & 1 \\ 0 & 6 & 2\end{array}\right]\)
\(\left[\begin{array}{rr}1+\mathrm{i} & 3 \\ 0 & \mathrm{i}\end{array}\right]\)
\(\left[\begin{array}{rrr}1 & -2 & 1 \\ -2 & 4 & -2 \\ 0 & 0 & 1\end{array}\right]\)
\(\left[\begin{array}{rrrr}2 & 1 & -1 & -2 \\ 0 & 1 & -1 & -2 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]\)
3
By hand, find a matrix \(P\) such that \(P^{-1} A P\) is diagonal for each of the following matrices. Then use \(P\) to find a formula for \(A^{k}, k\) a positive integer.
\(\left[\begin{array}{lll}2 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 3\end{array}\right]\)
\(\left[\begin{array}{lll}1 & 2 & 2 \\ 0 & 0 & 0 \\ 0 & 2 & 2\end{array}\right]\)
\(\left[\begin{array}{ll}1 & 2 \\ 3 & 2\end{array}\right]\)
\(\left[\begin{array}{ll}0 & 2 \\ 2 & 0\end{array}\right]\)
\(\left[\begin{array}{rrrr}2 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 1\end{array}\right]\)
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:
\(\left[\begin{array}{rr}-7 & -6 \\ 9 & 8\end{array}\right]\)
\(\frac{1}{3}\left[\begin{array}{ll}1 & 3 \\ 2 & 0\end{array}\right]\)
\(\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]\)
\(\frac{1}{2}\left[\begin{array}{lll}1 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 2 & 1\end{array}\right]\)
\(\left[\begin{array}{rr}1 & 1 \\ -1 & -1\end{array}\right]\)
6
If the matrices of the previous exercise are transition matrices, for which do all \(\mathbf{x}^{(k)}\) remain bounded as \(k \rightarrow \infty\) ? 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).)
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 \(50 \%\) 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).
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.
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?