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?
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.
Please help me!
/Maria
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 and are free
edit
After doing some more reading i realised that get the basis
=[-1,1,0]
=[-1,0,1]
But i cant figure out the basis for .
If i understand this correctyl i need 3 eigenvectors since the first matrix is 3x3.
I don't understand, should I put t and s as answers for and in the original matrix as such:
and then do a gaussian elimination? If so i get the following equations:
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:
This would give me the matrix
Normalizling this gave me:
Am I going the right way in any of these cases?
/ Maria
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
edit
but the first and second matrix should be within brackets
Lets see if I understand this correctly, to diagonolize my matrix (A) I need a diagonalizing matrix (P) and a Lambda*I matrix (D)
After finding my eigenvalues (1) I know that my D matrix is:
I also know that AP = PD
And that P has to be a nxn matrix
After doing all the calculations I have gotten
But from here i dont know how to move on, Are not and 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?
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.
Response to # 9:
Your matrix D will be as follows:
You construct your P from the orthonormalized eigenvectors. Let and be the eigenvectors corresponding to and let be the eigenvector corresponding to And let's assume that these are already-normalized eigenvectors. Then the matrix (or , you'll have to double-check) is given by