A similar question on finding the values for which its digonolized

[transgalactic]

http://img139.imageshack.us/my.php?image=img86081ca6.jpg

first i am given an all 1 matrix
and i was asked to find the eigenvalues
and the eigenvectors of this metrix

then i was told to find the orthonormal basis of this matrix
so i took the vectors that i have got
i made an gram shmidt formula on the two eigenvectors which came out of
the 0 eigenvalue and divided each vector of the resolt by its NORMA

after that i took the remainding third eigenvector and i divided him by his norma
is it ok??
(this is how i was told to solve this kind of question
but if a have only a simple matrix and i want to turn it into orthonormal
and i don't have such a relations between the columns as here(regarding the eigenvalues)
what should i do then??(this is only a subquestion not the main one)

then i was given an all "z" matrix
and i was told to proove that for each value of Z which belongs to the complex numbers "C" then its diagonizable
i don't know how to proove that

so if i put in z=0 then for this option i have only 2 vectors as i showed in the start
instead of the desiarble 3

so the answer that i think is z differs 0

i don't know how to proove what they want

 

[mighty2000]

Thank you for your question. I would like to offer some guidance on how to approach this problem.

First, let's define what eigenvalues and eigenvectors are. Eigenvalues are special numbers associated with a matrix that represent the scaling factor of the eigenvectors. Eigenvectors, on the other hand, are special vectors that when multiplied by the matrix, result in a scalar multiple of itself.

In order to find the eigenvalues and eigenvectors of a matrix, we can use the following steps:

1. Find the characteristic polynomial of the matrix by subtracting the eigenvalue from the diagonal entries of the matrix and taking the determinant.

2. Set the characteristic polynomial equal to 0 and solve for the eigenvalues.

3. For each eigenvalue, solve the system of equations (A - λI)x = 0 to find the corresponding eigenvector.

Now, for the orthonormal basis, we can use the Gram-Schmidt process to find a set of orthogonal vectors. However, in order to find an orthonormal basis, we need to normalize the orthogonal vectors by dividing them by their norms. This ensures that the vectors have a length of 1. So, in your case, if you have two orthogonal eigenvectors and one non-orthogonal eigenvector, you would need to normalize all three vectors to create an orthonormal basis.

As for the question about the "z" matrix, it is important to note that not all matrices are diagonalizable. A matrix is diagonalizable if it has a full set of linearly independent eigenvectors. In order to prove that a matrix is diagonalizable, we need to show that it has a full set of linearly independent eigenvectors. This means that for every eigenvalue, there is at least one corresponding eigenvector. In your case, if you are given an all "z" matrix, you would need to show that for every complex number z, there exists at least one eigenvector. This can be done by solving the system of equations (A - λI)x = 0 for each value of z.

I hope this helps guide you in solving this problem. If you have any further questions, please do not hesitate to ask. Good luck!