MATH221 April 2013
• Q1 • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q3 • Q4 • Q5 • Q6 • Q7 (a) • Q7 (b) • Q7 (c) • Q8 • Q9 • Q10 • Q11 • Q12 (a) • Q12 (b) • Q12 (c) •
Find a diagonal matrix D and an invertible matrix P such that . Do not compute the matrix . Show that approaches the zero matrix as k becomes very large.
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?
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You first need to compute the eigenvalues and eigenvectors for the matrix. How will you then package them together to find P and D?
For the second part, use that
Then expand and simplify the right hand side.
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To find the diagonal matrix D, we need to compute the determinant of the matrix A-tI.
Which means that eigenvalues are 3, 5 and -2. To find the eigenvectors (which will be the columns of the matrix P), we pick vectors of the kernels of where t = 3, 5, -2.
Eigenvalue t = 3.
Thus an eigenvector corresponding to the eigenvalue t = 3 is given by
Eigenvalue t = 5.
Thus an eigenvector corresponding to the eigenvalue t = 5 is given by
Eigenvalue t = -2.
Thus an eigenvector corresponding to the eigenvalue t = -2 is given by
- can be chosen as an eigenvector.
So the matrices D and P are given by
Now, as , then
So we can see that as k gets large, get closers to the zero matrix. Hence the right hand side, and with it the matrix A-k, also approaches the zeros matrix.