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There was a typo in my explanation of how to diagonalize a matrix A (finding D and P). It was minor. The technique I sent earlier was correct, but I accidentally wrote an equation wrong in my summing up, that I have repaired.
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Don't have my book or audio? You can download a free sample of my book and audio lectures containing Lessons 1 and 2: Did you read my tips on how to study and learn this course? If not, here is a link to those important suggestions: The department posted SOLUTIONS for Assignment 1 (these are not my solutions). Click here.The department posted SOLUTIONS for Assignment 2 (these are not my solutions). Click here.The department posted SOLUTIONS for Assignment 3 (these are not my solutions). Click here.Here is a link to the actual assignment, in case you don't have it: Study Lesson 15 (Linear Transformations) and Lesson 16 (Eigenvalues and Eigenvectors) from my Linear Algebra & Vector Geometry book to prepare for this
assignment.
Do note: Lessons 11 to 14 in my book are OMITTED. They have removed those topics from the course.
Of course, always seek out assistance from my book, your course notes, etc. if you ever hit a question you don't understand, but try not to be learning things as you do an assignment. Learn first, then put your learning to the test. Similar to my Lesson 15, #2 and #3.
Similar to my Lesson 15, #7.
Remember the two properties of Linear Transformations:
- T(u + v) = T(u) + T(v)
(You can split the
transformation of a sum of vectors into the sum of two separate transformations)
- T(cu) = cT(u)
(You can factor a scalar multiple c out of a linear transformation.)
Hint: If you know the transformations of the two elementary vectors e1 and e2, you can make the standard matrix. The information given allows you to determine T(e1) and
T(e2).
Similar to my Lesson 16, #3 and #4.
However, I do not discuss Diagonalization of Matrices in my lesson. Do make sure you know how to do this! It may be on your exam. It is really quite
simple.
Given a matrix A, that is size n x n: - Compute the eigenvalues of A. Let's call them λ1, λ2,...λn.
- Compute the corresponding eigenvectors. Let's call them v1, v2,...vn.
- Then, we can define a diagonal matrix D where D's entries down the main diagonal are λ1, λ2,...λn, respectively.
- We can also construct a matrix P where each column of P is the
associated eigenvector. So, if you put λ1 in the top left corner (the 1,1 position) of D, then you put its associated eigenvector v1 down the first column of P. If you put λ2 in the 2,2 position of D, then put its associated eigenvector v2 down the second column of P, etc.
- Then matrix A can be expressed as in the form A = PD(P-inverse) where D is a diagonal matrix consisting of the eigenvalues (step 3) and P is the matrix
consisting of the eigenvectors (step 4).
Why is this important? Because it enables us to compute powers of matrices quite quickly. As I showed back in Lesson 3, it is very easy to compute powers of a diagonal matrix. If you want to compute D^10 for example, you just raise each entry on the main diagonal to the power of 10. Whereas, if A is not a diagonal matrix, the only way to compute A^10 is to compute A*A*A*A... ten
times.
If A is diagonalizable. Which is to say, if A = PD(P-inverse), then it also follows that
A^n = PD^n (P-inverse). Which is to say, if you want to raise matrix A to any power n, then you merely have to raise its associated diagonal matrix D to the power of n, then multiply P by D^n by P-inverse to get the answer for A^n.
For example, in my Lesson 16, #3(a), I gave you the 2x2 matrix A, where A = [6 10] is the first row [-5 -9] is the second row
We found the eigenvalues λ= 1 and λ= -4. Therefore, we can state the diagonal matrix D= [1 0] is the first row [0 -4] is the second row (Which is to say, put λ= 1 in the (1,1) position and λ= -4 in the (2,2)
position.
Do note that you could just as easily put -4 in the (1,1) position and 1 in the (2,2) position. It doesn't matter which order you list the eigenvalues in the matrix D. What matters only is that, once you have decided which eigenvalue to put in (1,1), you must put its associated eigenvector v1 down the first column of P.
In my Lesson 16, #4(a) I found the associated
eigenvectors. - For λ= 1, I found the eigenvector to be (-2t, t). We need a specific eigenvector to make P, so simply sub in t=1. So, my associated eigenvector will be (-2,1). (I could sub in any nonzero value for t to make an eigenvector). That vector will be put down the first column of P.
- For λ= -4, I found the eigenvector to be (-t, t). We need a specific eigenvector to make P, so simply sub in t=1. So, my associated eigenvector will be
(-1,1). (I could sub in any nonzero value for t to make an eigenvector). That vector will be put down the second column of P.
Therefore, the matrix P associated with the diagonal matrix D, will have (-2,1) written down the first column, and (-1,1) written down the second column, so P= [-2 -1] is the first row [1 1] is the second row
Of course, I could now easily compute the inverse of P by using the shortcut
for 2x2 matrices. The determinant of P = -2-(-1) = -1, so P-inverse = [-1 -1] is the first row [1 2] is the second row I switched the order of the main diagonal of P, and changed the signs of the back diagonal, then multiplied by the reciprocal of the determinant of P, which is 1/-1 = -1.
So, that is what is required for you to answer question 3(c) in this assignment. Your answer to part (a), the
eigenvalues, will tell you the diagonal matrix D. Your answer to part (b), the eigenvectors, will tell you the matrix P.
In part (d), compute P-inverse using the shortcut for 2x2 matrices. Then literally compute the product P-inverse times D times P and confirm that you end up with the original matrix A to verify that A = PD(P-inverse).
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