Homework 3 collaboration area
Question from James Down Under (Jayling):
For Page 329 Question 11. Am I meant to calculate all eigenvalues and eigenvectors or just calculate the eigenvector corresponding to the given eigenvalue of 3?
Answer from Steve Bell :
Yes, you are only supposed to find the eigenvector for lambda=3. (The idea here is to spare you from finding the roots of a rather nasty 3rd degree polynomial.)
Jayling: thanks Steve, I did try the hard way first but then started to drown in the algebra.
Question from a student:
Let 3x+4y+2z = 0; 2x+5z= 0 be the system for which I have to find the basis.
When Row Reduced the above system gives [ 1 0 2.5 0 ; 0 1 -1.375 0].
Rank = no of non zero rows = 2 => Dim(rowspace) = 2 ; Nullity = # free variables = 1
Q1: Aren't [ 1 0 2.5] and [0 1 -1.375] called the basis of the system?
A1 from Steve Bell:
Those two vectors form a basis for the ROW SPACE.
The solution space is only 1 dimensional (since the number of free variables is only 1).
Q2: Why is that we get a basis by considering the free variable as some "parameter" and reducing further(and get 1 vector in this case). Isn't that the solution of the system?
A2 from Steve Bell :
If the system row reduces to
[ 1 0 2.5 0 ] [ 0 1 -1.375 0 ]
then z is the free variable. Let it be t. The top equation gives
x = -2.5 t
and the second equation gives
y = 1.375 t
and of course,
z = t.
So the general solution is
[ x ] [ -2.5 ] [ y ] = [ 1.375 ] t [ z ] [ 1 ]
Thus, you can find the solution from the row echelon matrix, but I wouldn't say that you can read it off from there -- not without practice, at least.
Question from a student :
On problem 11, I swapped rows 1 and 2 during row reduction and my final solution has x1 and x2 swapped. Do I need to swap back any row swaps or did I make a mistake along the way? Tlouvar
Eun Young discussed this issue here in a way that is slightly beyond the scope of our course, so I've moved it to here:
Remark from Steve Bell :
Step 1: Find the eigenvalues from det(A - lambda I)=0.
Step 2: Choose an eigenvalue lambda and plug it into the system
(A - lambda I) a = 0
and solve the system for the eigenvector a. Swapping rows does not change the answer, so you are safe here.
Sometimes you might think you are swapping entries of a vector when you are really multiplying by -1. For example , if [1, -1] is an eigenvector, so is [-1, 1].
Question from Dalec
For #2 on page 351, I found my spectrum to be lambda = 2i , and -i. For the case where lambda = 2i , I am trying to find the eigenvectors, and I get a matrix
[ -i 1+i | 0] [ -1+i -2i | 0]
Is there a way to get a 0 in the bottom left, or is this simply overcontrained?
- Chris
Suggestions from Shawn Whitman
In one step: multiply row 1 by (1+i) and add to row 2.
In two easier steps: Multiply row 1 by i,
[1, (-1+i)]
[(-1+i), -2i]
then multiply row 1 by (1-i) and add to row 2.
[1, (-1+i)]
[0, 0]
I have questions about determinants. For a homogeneous systems, for non-zero determinants we have only the trivial solution while for zero determinant we have infinitely many solutions. For non-homogeneous system, when the determinant is non-zero we have exactly one solution. 1. What will happen if a non-homogeneous system has zero determinant? 2. From the determinant of a non-homogeneous system can we know when the system doesn't have any solution?
- Farhan
Question from Ryan Russon:
About p. 338, #3,6, and 8, are we supposed to be finding eigenvectors here? I noticed that they put them in the back of the book, although it only asks to find the spectrum of each, which was defined as the set of eigenvalues in 8.1? I understand that we are using Thms 1-5 to prove our results and it seems like #3 doesn't require finding eigenvectors to prove that it isn't any of the listed matrices. I hope I am not way off-base here. Thanks!
Follow-up question: On p. 338, #6 Are we only to consider $ A \in \mathbb{R}^{n \times n} $ or are we to consider complex matrices as well? Thanks again!
Response from Jake Eppehimer:
I found that #8 is orthogonal, according to theorem 5. It took quite a bit of manipulation with trig identities, but I believe my answer is reasonable. For number 6, I am not exactly sure how to find the eigenvalues. I am considering substituting a couple prime numbers for k and a, but I am unsure if that is the correct way to do it.
