Homework 10

1. Diagonalization
2. Counterexamples
3. Square Root of a Matrix
4. Extinction (Programming)

The following assignment is due Thursday 11/21 by 11:59 PM. You should submit all your written solutions to Gradescope as a single pdf. Follow the instructions in the programming problem for code solutions.

Your solutions must be exceptionally neat and the final answer in your solution to each problem must be abundantly clear, e.g., surrounded in a very visible box. The graders have license to dock points for illegible or unclear solutions.
For the written part, choose the correct pages corresponding to each problem in Gradescope. Note that Gradescope registers your submission as soon as you submit it, so you don't need to rush to choose the pages. You will receive no credit if you do not choose the correct pages, no exceptions.

1. Diagonalization

For each of the following matrices, determine

its characteristic equation in factored form
a diagonalization, if possible

If the matrix is diagonalizable, then the diagonalization you give should have integer values in \(P\), \(D\) and \(P^{-1}\), and the entries in \(D\) should be non-increasing from left to right. If the matrix is not diagonalizable, write the matrix is not diagonalizable and justify your answer. You must show your work and do all calculations by hand.

1.1. (3 points)

\begin{align*} \begin{bmatrix} 9 & 4 \\ -14 & -6 \end{bmatrix} \end{align*}

1.2. (3 points)

\begin{align*} \begin{bmatrix} -6 & -7\\ 7 & 8 \end{bmatrix} \end{align*}

1.3. (4 points)

\begin{align*} \begin{bmatrix} 2 & 0 & 0 & 0 \\ -1 & 3 & 0 & 0 \\ 3 & -3 & 2 & 0 \\ -3 & 18 & -6 & -1 \end{bmatrix} \end{align*}

1.4. (4 points)

\begin{align*} \begin{bmatrix} -1 & -3 & 0 \\ 2 & 4 & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{align*}

Please only use tools that we have discussed in this course. This means you can use the equation for determinants of \(3 \times 3\) matrices or you can use row reductions.¹

2. Counterexamples

For each of the following statements, find a counterexample by giving explicit \(2 \times 2\) matrices \(A\) and \(B\) which falsify the statement. Justify your answer.

2.1. (3 points)

For any matrices \(A\) and \(B\) \(\det(A + B) = \det(A) + \det(B)\).

2.2. (3 points)

For any matrix \(A\), if \(\det(A - \lambda I) = (\lambda - 1)^2\), then \(\dim(\mathsf{Nul}(A - I)) = 2\).

2.3. (4 points)

For any matrices \(A\) and \(B\), if \(\det(A - \lambda I) = \det(B - \lambda I)\), then \(A\) is similar to \(B\). Hint: If \(A\) is diagonalizable and \(A\) is similar to \(B\) then \(B\) is also diagonalizable.

3. Square Root of a Matrix

(5 points) Consider the matrix

\begin{align*} A = \begin{bmatrix} 103 & -8 & -47 \\ 24 & 1 & -12 \\ 198 & -16 & -90 \end{bmatrix} \end{align*}

Determine a matrix \(B\) such that \(B^2 = A\). Hint. \(A\) is diagonalizable. You can use can use a computer to find eigenvectors (e.g., np.linalg.eig) or to reduce matrices, but you must otherwise show your work. Justify your answer.

4. Extinction (Programming)

(16 points) Suppose you're working as a wildlife biologist and you've recently discovered a new species of seabird living on a remote island in the Aegean Sea (congratulations). You've been gathering data on the survival rates of these birds and have determined the following.

X percent of CHICKs survive to become ADOLESCENT birds each year
Y percent of ADOLESCENT birds survive to become ADULT birds each year
ADOLESCENT birds do not lay any eggs
Z percent of ADULT birds survive each year
ADULT birds succeed in hatching N CHICKs per pair of ADULT birds each year

This situation can be described as a linear dynamical system. From this information you're interested in the conditions under which this species of bird is expected to go extinct. Assuming this (over-)simplified system, it suffices to check if the largest eigenvalue of the underlying matrix is less than 1 (make sure you understand why this is the case). In this problem, you will be writing several functions to gauge how the above parameters affect the possible extinction of this species of seabirds.

You are given starter code in the file hw10.py. Don't change the name of this file when you submit. Also don't change the names of any functions included in the starter code. The only changes you should make are to fill in the provided TODO items. You are required to implement the following functions.

survival_matrix, which defines the evolution matrix for the above situation
min_non_extinct_rate_chicks, which determines the smallest survival rate of chicks required to ensure non-extinction of the species
min_non_extinct_rate, which determines the smallest survival rate of adolescents and adults for which non-extinction is possible
years_until_extinct, which determines the number of years it will take for a particular population to go extinct (and None if the species is not expected to go extinct)

See the docstrings in the starter code for more details. You'll upload a single file hw10.py to Gradescope, where you can verify that it passes some (but not all) autograder tests.

Footnotes:

But you cannot, for example, use any definition of determinants that depends on cofactors.