MATLAB Linear Algebra assignment (matrices/eigenvalues)

Introduction to MATLAB and Leslie Matrices

1) Start MATLAB

2) You can type commands directly into the command window: Type and expression and then hit enter to evaluate the expression. For example you can add 2 and 2:

>> 2+,2

If you want to suppress the output of a command follow it with ‘;”. For example “2+2”;

3) Practice evaluating a few expressions in the command window. (In MATLAB multiplication is represented by * so 3*2=6).

4) You can also define variables in the command window. For example, to set the value of the variable x equal to 2 just type “x=2” and hit enter. Notice that the variable x now appears in your workspace on the far right hand side.

5) Variables can also be vectors or matrices. For example “x=[1 2 3]” creates a row vector with x(1)=1, x(2)=2, and x(3)=3. What happens when you type “x=[1;2;3]”?

6) Create a 3x3 matrix named A with rows [1 2 3], [4, 5, 6], and [7, 8, 9].

7) You can also create functions in the command window. We will create functions of two variables, x and y. (Later these functions will be the right hand side of a first order differential equation.) For example to create a function g(A,x)=A*x type “g=@(A,x) (A*x)” and press enter.

8) Now that you have created the function g(x,y) you can evaluate it at different x and y values. For example, to compute g(1,2), just enter: “g(1,2)”. Also compute g(A,x0), for x0=[1 1 1]T. Note that x0 must be passed to g as a column vector so that the dimensions of A and x0 are consistent for multiplication.

9) To can use a for loop to compute and store the values of Anx for several values of n. For example, type “y=zeros(3,4); for i=1:4 y(:,i)=A^i*x; end” to store the values of An for n=1, . . , 4 as the columns of the matrix y. That is the ith column of y is Aix0.

10) You can plot your answer by typing “plot([1 2 3 4], y, ‘o’)”.

11) To compute the eigenvalues and eigenvectors of a matrix A in MATLAB, type “[V, D]=eig(A)”.

The matrix V has the eigenvectors of A as its columns. The matrix D has the corresponding eigenvalues of A on its diagonal. Find the eigenvalues and eigenvectors of the matrix A that is stored in your workspace.

12) If x is a vector, the command “sum(x)” returns the sum of the elements in x. Normalize the eigenvectors of A so that each eigenvector sums to one.

13) Construct the Leslie matrix for the northern fur seal population in exercise 22 on page 359 of your text.

14) Find the eigenvalues and eigenvectors of this Leslie matrix. Predict the long-term population growth rate and steady state population structure.

15) Suppose that a group of fur seals, including 50 females between the ages of 2 and 4 years are introduced to a large island off the coast of British Columbia. Find the total number of female seals predicted to be in the population 1-10 years later. Plot your answer. Edit your plot so that it is easy to read and includes a title and x- and y-axis labels.

16) Predict the percentage of seals aged 0-2 years in years 0-10 (even years only). Plot your answer. Edit your plot so that it is easy to read and includes a title and x- and y-axis labels.