Question

Interpret the matrix as linear transformation on C^3, find a basis of C^3 such that the matrix of the transformation with respect to that basis is triangular
\[\left[\begin{matrix}0&1&0\\0&0&1\\1&0&0\end{matrix}\right]\]
This is my stuff and where I stuck

Answer 1

isn't it permutation matrix?

Answer 2

what is C^3 ??

Answer 3

yes, I think so but I have to do steps like this. My prof said that the goal of the exercise is students know how to find the triangular matrix on this way.
by observing, we can se e1--> e2-->e3-->e1.

Answer 4

@experimentX ignore it, it's standard basis like R^3

Answer 5

\[
T(e_1)=e_3\\
T(e_2)=e_1\\
T(e_3)=e_2
\]

Answer 6

In general
\[

T(x,y,z)=(y,z,x)
\]

Answer 7

Yes, I meant that

Answer 8

then?

Answer 9

I gave you the transformation and that is it

Answer 10

(1, 1, 1) is invariant under transformation. what are M2 and M3 ??

Answer 11

\[T=\left[\begin{matrix}0&0&1\\0&1&0\\1&0&0\end{matrix}\right]\]but it is not triangular form @eliassaab

Answer 12

@experimentX M1= R(1,1,1) is invariant under the transformation
and I found it out as I post, the attachment has 2 pages, please, scroll it down

Answer 13

The way I'm understanding this problem, is that the easiest way to do it is to find the eigenvalues/eigenvectors. If the eigenvalues are distinct, then the eigenvectors will be linearly independent, and will thus give a basis. Furthermore, the matrix with respect to the basis formed by the eigenvectors will be diagonal, and thus triangular.

Answer 14

i was thinking the same
http://www.wolframalpha.com/input/?i=Eigensystem%5BTranspose%5B%7B%7B0%2C+1%2C+0%7D%2C+%7B0%2C+0%2C+1%7D%2C+%7B1%2C+0%2C+0%7D%7D%5D%5D

Answer 15

*similar ... lol

Answer 16

i read your pdf, but can't figure it out what are M2 and M3.

Answer 17

That's the problem. :) if I go to this way, I will stuck at complex eigenvalue because it will give a mess!! That's why we, students, study this way.

Answer 18

okay, you found out M1, M2 and M3 ... so far I know for a vector, change of basis means some transformation, ,,, how do you change matrix is basis is changed?

Answer 19

M1: 1-D
M2: 2-D
M3:3-D