Science:Math Exam Resources/Courses/MATH307/April 2005/Question 07
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Determine whether or not the following two matrices are similar:
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!
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.
We are going to work under the assumption that the two matrices are similar and use properties of similar matrices to aid us in solving this question.
For two matrices to be similar and must satisfy the equation
The main properties of similar matrices which we will use to verify this equation is that:
1) Similar matrices have the same eigenvalues 2) Similar matrices have the same Jordan canonical form
So, we need to get the two matrices into Jordan form and compare them.
The Jordan canonical form is given by
Where is just the matrix of eigenvectors.
We’re going to start with as it’s easy to pick out the eigenvalues from the diagonal, which must also be the eigenvalues of from our known properties.
So let’s get our eigenvectors!
You should know how to get eigenvectors from eigenvalues (solve )
The eigenvectors of are:
Besides knowing the formula for obtaining the inverse of a x matrix, one can also obtain it by ’rref’-ing the augmented matrix:
However you want to do it, we obtain:
Using this information we find that our J, the form we want to get, is:
Now we just do the same steps again for and then we’re done!
So, , and we already know our eigenvalues as they are just the same as B!
The eigenvectors of are:
And so and with we obtain
With these two matrices known we can see that:
And so the two matrices have the same Jordan canonical form, and are thus similar because of the following:
With us knowing that the two matrices have the same Jordan form we can show what and will be:
if and then
substituting that into our other equation gets:
So just let and
and we get which is what we sought out to begin with!