Science:Math Exam Resources/Courses/MATH152/April 2017/Question A 25

MATH152 April 2017

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Question A 25
A matrix $A$ is entered into MATLAB. The eigenanalysis of $A$ is performed using the command [T D] = eig(A) which gives the following results: T = 0.5257 0.0995 0.8507 0.9950 D = 1.0000 0 0 3.0000 Using these results, determine $A^{3}[1,10]^{T}$. .

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

It is important to know how the eig command works. The matrix T consists of the eigenvectors of A, normalized so that each eigenvector has length 1. Thus A has the diagonalization $A=TDT^{-1}.$ One way to solve this problem would be to compute $A^{3}$ directly, but this is laborious and the time given for the exam is limited. Take a careful look at the matrix T and see what is special about the vector $\left({\begin{array}{c}1\\10\end{array}}\right).$

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The eig() command returns the eigenvalues and eigenvectors of the matrix A. Therefore, the columns of T are the eigenvectors of A, normalized to have length 1, and the eigenvectors of A are 1 and 3. It is possible to compute the matrix $A^{3}$ using the formula $A^{3}=TD^{3}T^{-1}$ but this is impractical given the time constraints for the exam. This can be circumvented by carefully considering the eigenvectors of A. Once such eigenvector is given by $\left({\begin{array}{c}0.0995\\0.9950\end{array}}\right).$ The key to this problem is to notice that this vector is a scalar multiple of our input vector $\left({\begin{array}{c}1\\10\end{array}}\right).$ Therefore, the input vector is, in fact, an eigenvector of A with eigenvalue 3 (because the diagonal entry in D corresponding to this eigenvector is 3). Therefore, multiplying by the matrix A is equivalent to simply multiplying the components of our vector by 3. Because we multiply by A three times, we get $A^{3}\left({\begin{array}{c}1\\10\end{array}}\right)=\left({\begin{array}{c}27\\270\end{array}}\right).$ Answer: $\color {blue}\left({\begin{array}{c}27\\270\end{array}}\right).$