MATH152 April 2017
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Question A 25
A matrix is entered into MATLAB. The eigenanalysis of is performed using the
command [T D] = eig(A) which gives the following results:
Using these results, determine .
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!
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
One way to solve this problem would be to compute 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
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.
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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 using the formula
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
The key to this problem is to notice that this vector is a scalar multiple of our input vector
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