Introduction to MATLAB Exercise 1

Setup MATLAB Directory

• Create a directory "MATLAB" in your P-drive, and put all your future matlab files in it (create subdirectories if needed to organize the files better).
• Better to work with matlab files (save it as "lab1.m" first) instead of typing in the command window, so that you can always modify the commands later.
• Only keep the "editor" for you to type commands and "command window" to see the output, and you do not need other windows (like "workspace") most of the time.
• You can separate different block of code using double percent symbol %% Question 1 (also serve as a line of comment), to avoid excessive output from commands in the file.
• Use keyboard shortcuts like Ctrl+Enter(execute the current cell) to improve your productivity, instead of clicking "run" with your mouse every time.
• Do not be afraid of making mistakes (you are not doing any damage to the computer)! You may get a huge amount of annoying error messages, because of the spelling mistake in the command, inappropriate operations, unclosed bracket. So try and error, or get the lecturer or the demonstrators to help you debug the code.

Exercises in MATLAB

1. Create the following 10-by-10 matrix, with 8-by-8 submatrix (all ones) and zeros around it: \begin{equation} A = \begin{bmatrix} 0 & 0 & 0 & \cdots & 0 & 0 & 0 \cr 0 & 1 & 1 & \cdots & 1 & 1 & 0 \cr 0 & 1 & 1 & \cdots & 1 & 1 & 0 \cr \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \cr 0 & 1 & 1 & \cdots & 1 & 1 & 0 \cr 0 & 1 & 1 & \cdots & 1 & 1 & 0 \cr 0 & 0 & 0 & \cdots & 0 & 0 & 0 \end{bmatrix}. \end{equation}
2. Create a 3-by-3 rand matrix using rand and find its eigenvalues and the associated eigenvectors using eig. Verify directly the definition \(Ax=\lambda x\) by extracting the eigenvalue and eigenvector from your output.
3. Solve the following system using x = A\b (define the matrix A and the colum vector b first): \begin{align} 3x+4y+5z &=2 \cr 2x-3y+7z &=-1\cr x-6y+z &=3. \end{align} Does Ax is exactly b (look at the difference A*x-b)?
4. The Hilbert matrix of order \(n\) is defined as \begin{equation} A = \begin{bmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n} \cr \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \cdots & \frac{1}{n+1} \cr \vdots & \vdots & \vdots & \ddots & \vdots \cr \frac{1}{n} & \frac{1}{n+1} & \frac{1}{n+2} & \cdots & \frac{1}{2n-1} \end{bmatrix}, \end{equation} that is the entry \(a_{ij}=1/(i+j-1)\). You can generate this special matrix using hilb(n). Let A be the Hilbert matrix of order \(11\) and x be a random vector of length \(11\). Define b=A*x, and xx=A\b. What do you expect the size of x-xx? Try the same with a random matrix A of size \(11\).
5. Calculate \(A^{1000}\) and \(A^{-10}\), where \begin{equation} A = \begin{bmatrix} 3/2 & 1/2 \cr -1/2 & 1/2 \end{bmatrix}. \end{equation} Check whether you get the exact answer \begin{equation} A^{1000} = \begin{bmatrix} 501 & 500 \cr -500 & -499 \end{bmatrix},\qquad A^{-10} = \begin{bmatrix} -4 & -5 \cr 5 & 6 \end{bmatrix}, \end{equation} by taking the difference between the numerical output and the above matrices.
6. You can get the diagonal element of a matrix A by using diag(A). For a vector a, you can also create a matrix with diagonal element a using diag(a). Experiment the following commands:

       A = magic(3)
       diag(A)
       a = [1 2 3]
       diag(a)

   Does it matter whether a is a column or row matrix? Try to apply diag to the transpose of a, which is simply a'.
7. You can retrieve the lower and upper triangular part of a matrix using tril and triu. Create a 5-by-5 random matrix A and check the output of tril(A,2) , tril(A,-1) , triu(A,3) and triu(A,-2) .
8. Based on the experiments from the last question, use tril and triu to retrieve the tri-diagonal part (the diagonal, with two sub-diagonal right above and below the main diagonal) of a matrix: \begin{equation} \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \cr a_{2,1} & a_{2,2} & \cdots & a_{2,n} \cr \vdots & \vdots & \ddots & \vdots \cr a_{n,1} & a_{n,2} & \cdots & a_{n,n} \end{bmatrix} \qquad \to \qquad \begin{bmatrix} a_{1,1} & a_{1,2} & 0 & \cdots & 0 \cr a_{2,1} & a_{2,2} & a_{2,3} & \cdots & 0 \cr 0 & a_{3,2} & a_{3,3} & \cdots & 0 \cr \vdots & \vdots & \vdots & \ddots & \vdots \cr 0 & 0 & 0 & \cdots & a_{n,n} \end{bmatrix}. \end{equation} Try your command to find the 10-by-10 tridiagonal matrix with ones \begin{equation} \begin{bmatrix} 1 & 1 & 0 & \cdots & 0 & 0 & 0\cr 1 & 1 & 1 & \cdots & 0 & 0 & 0\cr 0 & 1 & 1 & \cdots & 0 & 0 & 0\cr \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots\cr 0 & 0 & 0 & \cdots & 1 & 1 & 0 \cr 0 & 0 & 0 & \cdots & 1 & 1 & 1 \cr 0 & 0 & 0 & \cdots & 0 & 1 & 1 \end{bmatrix}. \end{equation}
9. Verify numerically that the determinant of two matrices is the product of these two matrices (that is \(\mbox{det}(AB)=\mbox{det}(A)\mbox{det}(B)\)) for two random 5-by-5 matrices (either uniform random rand or normal random randn), using det for the determinant of a matrix. (Here "verify numerically" means that the numerical value of \(\mbox{det}(AB)-\mbox{det}(A)\mbox{det}(B)\) should be small, of order \(10^{-14}\) or smaller).
10. The characteristic polynomial of a n-by-n matrix is defined as \( P(\lambda)=\mbox{det}(\lambda I-A)\). You can get the polynomial using poly(A), whose output is the coefficients of the polynomial. For example, if A is the matrix 2*eye(3), then \begin{equation} P(\lambda) = \mbox{det}\begin{pmatrix} \lambda-2 & 0 & 0 \cr 0 & \lambda-2 & 0 \cr 0 & 0 & \lambda-2\end{pmatrix} =\lambda^3-6\lambda^2+12\lambda-8. \end{equation} The output from poly(2*eye(3)) is the vector [1 -6 12 -8]. Verify numerically the following statements for a 5-by-5 random matrix, starting with A = rand(5); p = poly(A);:
    • The coefficient of \(\lambda^4\), that is p(2), is the trace -trace(A).
    • The constant term, that is p(6), is the determinant -det(A).
    • The matrix satisfies its own characteristic polynomial numerically, that is \( P(A)=0\), using polyvalm (that is evaluate the polynomial with matrix argument. To evaluate a polynomial at a point or a vector, use polyval).