1. ## Finding diagonalize matrix

I need help finding an orthogonal matrix P which diagonalize the matrix M6

Matrix M6:=
17 16 16
16 17 16
16 16 17

I think that the eigenvalues of the matrix are 1 and 49/33 but i'm not sure.
Also i don't know how to continue from this.

/Maria

2. I would agree with the 1 as an eigenvalue. However, the 49/33 is incorrect. To find the orthogonal matrix P, you'll need to orthonormalize the eigenvectors via Gram-Schmidt. What do you get when you do that?

3. The problem is that i dont understand how to get the eigenvectors. If 1 is the eigenvalue of every row in the matrix i get

A - (lambda)*I gives the following matrix

16 16 16
16 16 16
16 16 16

Row reducing this matrix gives me

1 1 1
0 0 0
0 0 0

From here I dont know how to find the eigenvectors since both $\displaystyle x_2$ and $\displaystyle x_3$ are free

edit

After doing some more reading i realised that get the basis

$\displaystyle x_2$ =[-1,1,0]
$\displaystyle x_3$=[-1,0,1]

But i cant figure out the basis for $\displaystyle x_1$.

If i understand this correctyl i need 3 eigenvectors since the first matrix is 3x3.

4. Well, there may be two eigenvectors. Set $\displaystyle x_{2}=s$ and $\displaystyle x_{3}=t,$ and solve for all three components in terms of those two parameters, just like you normally do in Gaussian elimination with back substitution. What do you get?

5. I don't understand, should I put t and s as answers for $\displaystyle x_2$ and $\displaystyle x_3$in the original matrix as such:

$\displaystyle \begin{bmatrix}17&16&16&0\\16&17&16&s\\16&16&17&s\ end{bmatrix}$

and then do a gaussian elimination? If so i get the following equations:
$\displaystyle x_{1}+\dfrac{16}{17}x_{2}+\dfrac{16}{17}x_{3}}=0$

$\displaystyle x_{2}+\dfrac{16}{17}x_{3}=\dfrac{17s}{33}}$

$\displaystyle x_{3}=\dfrac{33t}{49}-\dfrac{16s}{49}}$

From these equations I am not sure how to move on to find the eigenvectors.

My first a attempts before using s and t gave me the following equations:

$\displaystyle x=\begin{bmatrix}x{_1}\\x{_2}\\x{_3}\end{bmatrix}= x_{2}\begin{bmatrix}-1\\x1\\x0\end{bmatrix}x_{3}\begin{bmatrix}-1\\0\\1\end{bmatrix}$

This would give me the matrix $\displaystyle \begin{bmatrix}-1&-1\\1&0\\0&1\end{bmatrix}$

Normalizling this gave me: $\displaystyle \begin{bmatrix}-1&\dfrac{-1}{2}\\1&\dfrac{-1}{2}}\\0&1\end{bmatrix}$

Am I going the right way in any of these cases?

/ Maria

6. No, to find the eigenvectors $\displaystyle x_{\lambda}$ corresponding to a particular eigenvalue $\displaystyle \lambda$ (you always have to find an eigenvector relative to a particular eigenvalue), you solve the linear system $\displaystyle (A-\lambda I)x_{\lambda}=0$ for $\displaystyle x_{\lambda}.$ What do you get when you do that? (This is the system in post # 3.)

7. I'm not sure I understand what you are saying, sorry english isn't my first language.

Should I substract the identity matrix multiplied with my eigenvalues and then multiply this matrix with the eigenvectors

$\displaystyle \begin{bmatrix}17&16&16\\16&17&16\\16&16&17\end{bm atrix} - \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix} * \begin{bmatrix}x_{\lambda}\\x_{\lambda}\\x_{\lambd a}\end{bmatrix}=0$

edit

but the first and second matrix should be within brackets

8. Kind of. Solve this system:

$\displaystyle \begin{bmatrix}16&16&16\\16&16&16\\16&16&16\end{bm atrix}\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatr ix}=\begin{bmatrix}0\\0\\0\end{bmatrix}.$

This corresponds to $\displaystyle \lambda=1.$ You get it down to

$\displaystyle \begin{bmatrix}1&1&1\\0&0&0\\0&0&0\end{bmatrix}\be gin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}=\begi n{bmatrix}0\\0\\0\end{bmatrix}.$

Set $\displaystyle x_{2}=s, x_{3}=t$. Then $\displaystyle x_{1}=-s-t,$ and you have

$\displaystyle \begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}=s\ begin{bmatrix}-1\\1\\0\end{bmatrix}+t\begin{bmatrix}-1\\0\\1\end{bmatrix}.$

Therefore, your eigenvectors corresponding to $\displaystyle \lambda=1$ are what?

9. Lets see if I understand this correctly, to diagonolize my matrix (A) I need a diagonalizing matrix (P) and a Lambda*I matrix (D)

$\displaystyle A=PDP^{1}$

After finding my eigenvalues (1) I know that my D matrix is:
$\displaystyle \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$

I also know that AP = PD
And that P has to be a nxn matrix

After doing all the calculations I have gotten
$\displaystyle x=\begin{bmatrix}x{_1}\\x{_2}\\x{_3}\end{bmatrix}= x_{2}\begin{bmatrix}-1\\x1\\x0\end{bmatrix}x_{3}\begin{bmatrix}-1\\0\\1\end{bmatrix}$

But from here i dont know how to move on, Are not $\displaystyle x_{}2$ and $\displaystyle x_{3}$ my eigenvectors?

And dont i need a third eigenvetctor?

After all this, i know that my matrix has to be orthonormalized, can I orthonormalize my matrix after i have found it or should i do this with my eigenvectors?

10. In order that the matrix be diagonalizable, there must be 3 independent eigenvectors. If the only eigenvalue for this matrix is 1 and there are only two independent eigenvectors, then the matrix is NOT diagonalizable. In order that this matrix be orthogonalizable, there must be either another eigenvector corresponding to eigenvalue 1, independent of the two you have found, or there must be another eigenvalue.

11. Right. In the OP, there was another eigenvalue mentioned: 49/33. So you get two eigenvectors from the eigenvalue 1, and one eigenvector from the eigenvalue 49/33.

12. Response to # 9:

Your matrix D will be as follows:

$\displaystyle D=\begin{bmatrix}1&0&0\\0&1&0\\0&0&49/33\end{bmatrix}.$

You construct your P from the orthonormalized eigenvectors. Let $\displaystyle \mathbf{x}_{1}$ and $\displaystyle \mathbf{y}_{1}$ be the eigenvectors corresponding to $\displaystyle \lambda=1,$ and let $\displaystyle \mathbf{z}_{49/33}$ be the eigenvector corresponding to $\displaystyle \lambda=49/33.$ And let's assume that these are already-normalized eigenvectors. Then the matrix $\displaystyle P$ (or $\displaystyle P^{-1}$, you'll have to double-check) is given by

$\displaystyle P=[\mathbf{x}_{1}\;\mathbf{y}_{1}\;\mathbf{z}_{49/33}].$

13. ## eigenvalues

So you mean that i dont have 1 as the only eigenvalue?
Because 49/33 can be eliminated to 1 with gauss-elimination too.

I am also a bit confused since you wrote in post #2 that 49/33 was not an eigenvalue.

14. Whoops. Maybe it would help if I read my own posts, sometimes! Well, you tell me. What are the eigenvalues? We have 1 as an eigenvalue, of algebraic and geometric multiplicity 2.

15. Again I don't quite follow you since my english isn't so good, does of algebraic and geometric multiplicity 2 mean that I can multiply the 1 and use a multiple of it?

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