Homework 4 collaboration area
Question from Ryan Russon
Ok, what in the world are we supposed to do with p. 345 #1? From what I gather, The problem asks us to find the eigenvectors of this matrix P, A (the given matrix), and then some new 'A' where A = P-1AP and show that x = Py, where y is an eigenvector of P and x is an eigenvector of both A and this new A. The eigenvalues of P are really crummy... are we allowed to use a calculator or MATLAB or something to help us with this one? Maybe I am just over-complicating this problem... HELP!
I could be mistaken, but I think question #1 should read "if y is an eigenvector of 'A' , show that x = Py are eigenvectors of A". Theorem 3 on p. 340 says that "if x is an eigenvector of A, then y = P-1x is an eigenvector of 'A' corresponding to the same eigenvalue". If that is the case, x = Py, and the only calculation you need to do on P is to find P-1 for 'A' = P-1AP. --Andrew Sauseda 23:22, 9 September 2013 (UTC)
from James Ayling: I agree with you Andrew
from Ryan Russon:
That makes A LOT more sense. Thanks guys!
from Ryan Leemhuis:
Glad I wasn't the only one to come here with this question. I agree...error in the book. This was realized after calculating the eigenvalues of P and finding they are not very "user friendly"
from James Ayling: Any suggestions on how to go about answering Page 345 Questions 24 and 25?
Answer from Eun Young :
Q. 24. By thm 4, A = X'D'X − 1 where X − 1 = XT.
If we set XTx = y, then we have Q = yTD'y. We just transformed Q to the canonical form. See P.343.
So, $ Q = x^T A x = y^T D y = \lambda_1 y_1^2 + \cdots + \lambda_n y_n^2 $ where XTx = y.
Hence, the values of Q are controlled by the sings of the eigenvalues.
We know that there is a one-to-one correspondence between all nonzero x and all nonzero y since X is invertible.
Hence, if Q > 0 for all nonzero x, Q > 0 for all nonzero y.
Consider y = ( 1 0 0 ... ). Then, the first eigenvalue should be positive.
Consider y = ( 0 1 0 ...). Then, the second eigenvalue should be positive.
In this manner, all eigenvalues are positive.
Conversely, if all eigenvalues are positive, Q >0 for all nonzero y. So, Q >0 for all nonzero x.
You can show the others similarly.
Q.25. Prob22 => $ 4x_1^2 + 12 x_1 x_2 + 13 x_2^2 = 16 $ <=> Q = xTA'xwhere $ A = \begin{bmatrix} 4 \ \ 6 \\ 6 \ \ 13 \end{bmatrix} $.
To show that this is positive definite, you need to check if a11 > 0 and d'e't(A) > 0.
From James Ayling: Thanks Eun for the clarification.
Question From bminchhoff: for problem 25 I understand that all principle minors must be positive for positive definiteness but how do we know if principle minors not positive are negative definite or indefinite?
From - Kunal Bansal: Sec: 8.4, Ques - 25: To show whether its negative definite or indefinite (for prob - 23)- I think Calculating eigen values will help. As one of the eigen value in this case is positive and other one is negative, that satisfies the condition for indefinite.
Hi all. I am facing a problem with Question 18, Section 4.3 (mixing problem). The way I understand it, the pure water entry of 12 gal/min into T1 should not reflect into the fertilizer rate change equations? Ofcourse it keeps the volume constant in both tanks. If I continue in this way, I find that the eigen vectors to the two Eigen values are identical. When I go to solve for the initial value conditions I am not able to proceed in getting values for the constants (getting c1+c2=-100, 2*c1+2*c2=200). I am getting stuck, please advise. Thanks! &&&
Response from Jayling: Yeah I was going a bit cross eyed too with this problem. However I did not get identical eigenvalues. Can you tell me what your fertiliser rate equations for T1 and T2 are? For me when you just consider the "fertiliser" the rate equations I get are:
y1'=-(16/200)*y1+(4/200)*y2
y2'=+(16/200)*y1-(4/200)*y2
The eigenvalues and eigenvectors I get from this are 0, -0.1 and [0.25 1]$ ^T $, [-1 1]$ ^T $.
When I solve for the initial conditions I get C1=-40 and C2=240.
The issue that I have with this is similar to the above person's issue: The fact that the amount of pure water rate going into the system is identical to the amount of mixed water rate leaving the system, does this imply that we can treat the fertiliser system as closed? Or should we consider a pure water rate such as y3' entering the system, and then instead you are solving a 3x3 system rather than a 2x2 system? Or am I as usual overthinking this?
Question from student: I found a general formula for principal minors of a 3X3 matrix. Is there a general formula for the principal minors or a 2X2 matrix?
