My Math Forum NEED MATRICES (EIGENVALUE HELP) ASAP

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November 23rd, 2012, 02:12 AM   #1
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NEED MATRICES (EIGENVALUE HELP) ASAP

Hi,
Okay I have absolutely no idea what to do here so i need your help asap! Ive been given homework with a deadline on matrices (I can try to hand in a day late) I honestly have no idea how to do it.

I have attatched it to this post in pdf.

If you can show me how to do these questions I would really appreciate it.

Possibly if u can also take a screenshot or photo of the steps you took to work them out. That would be awesome! Im really desperate.
Attached Files
 emat.pdf (23.6 KB, 11 views)

 November 23rd, 2012, 07:15 AM #2 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: NEED MATRICES (EIGENVALUE HELP) ASAP Are you saying that you do not know what the "characteristic polynomial" of a matrix is? For this matrix, it is given by the determinant: $\left|\begin{array}{ccc}-\frac{3}{2}- x & \frac{1}{2} & \frac{5}{2} \\ -3 & 2- x & 3 \\ -\frac{1}{2} & \frac{1}{2} & \frac{1}{2}- x\end{array}\right|$ Can you find that determinant?
 November 23rd, 2012, 09:46 AM #3 Newbie   Joined: Nov 2012 Posts: 5 Thanks: 0 Re: NEED MATRICES (EIGENVALUE HELP) ASAP I've managed to get the first one but the rest of the questions are killing me. lol I'm not too good at matrices because I haven't had time to really sit down and go through it so a lot of the stuff is really confusing. I am basically blank in everything here(except number 1 by luck) so can you do and explain EVERYTHING please... Thanks
 November 23rd, 2012, 09:48 AM #4 Newbie   Joined: Nov 2012 Posts: 5 Thanks: 0 Re: NEED MATRICES (EIGENVALUE HELP) ASAP Maybe make a youtube video even?
 November 23rd, 2012, 10:26 AM #5 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: NEED MATRICES (EIGENVALUE HELP) ASAP To find the eigenvalues, as in problem 1, solve the equation $-x^3+ 2x^2+ x- 2= 0$. That should be easy- you are told that -1 is an eigenvalue: $-1^3+ 2(1^2)+ 1- 2= 0$. I am confused by your saying that you have no idea how to do problems 2 or 3. Do you not know what eigenvectors and eigenvalues are? The definition of "eigenvalue" and "eigenvector" is that v is an eigenvector of A with eigenvalue $\lambda$ if and only if $Av= \lambda v$. Since you are told that 1 is an eigenvalue, a corresponding eigenvector must satisfy $\begin{bmatrix}-\frac{3}{2} & \frac{1}{2} & \frac{5}{2} \\ -3 & 2 & 3 \\ -\frac{1}{2} & \frac{1}{2} & \frac{1}{2}\end{bmatrix}$$\begin{bmatrix} x \\ y \\ z \end{bmatrix}$$= \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ Multiplying that out gives you three linear equations to solve for x, y, and z.
 November 25th, 2012, 01:14 PM #6 Newbie   Joined: Nov 2012 Posts: 5 Thanks: 0 Re: NEED MATRICES (EIGENVALUE HELP) ASAP Oh... I didnt know what to equate them to. is the answer -6x +2y +6z ? before the unit vector part..
 November 25th, 2012, 01:31 PM #7 Newbie   Joined: Nov 2012 Posts: 5 Thanks: 0 Re: NEED MATRICES (EIGENVALUE HELP) ASAP What about Q3?

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