where I is identity matrix
Chatgpt gave me an answer but I am interested in methods of doing that.
#How to find matrix A (3x3) which isn't diagonalizable and A^3 = I?
grand nymph
static salmonBOT
formal heath
Let A be any 3x3 matrix. Then it has a Jordan normal form decomposition A = PJP^-1. Now I = A^3 = PJ^3P^-1 which implies J^3 = I, So all this boils down to finding a Jordan block matrix whose cube is I. We don't want it to be a diagonal so we have three possibilities, two of which are essentially the same. First one is [a, 1, 0; 0, a, 1; 0, 0, a] and the second and third are [a, 1, 0; 0, a, 0; 0, 0, b] and [a, 0, 0; 0, b, 1; 0, 0, b]. However, if you raise any of these to the third power then you will find that they can never be the identity matrix for any choice of a and b. Therefore no such matrix J exists and hence no such matrix A exists
What did chatgpt give you? I just tried chatgpt and it gave me a wrong answer.
Chatgpt will give you wrong answers like this all the time and you need to check if what it gives you is valid if you are going to use it
formal heath
This also generalizes nicely if you replace the power 3 by anything positive and size of the matrix 3x3 by any nxn
floral nova
Let A be any 3x3 matrix. Then it has a Jordan normal form decomposition A = PJP^...
how do you deduce that J^3?
floral nova
where I is identity matrix
Chatgpt gave me an answer but I am interested in meth...
an example would be taking a matrix associated to a rotation of 120º
Consider the matrix A=[1 0 0; 0 -0.5 -sqrt(3)/2; 0 sqrt(3)/2 -0.5]
Or A=[-0.5 -sqrt(3)/2; sqrt(3)/2 -0.5]
floral nova
Or A=[-0.5 -sqrt(3)/2; sqrt(3)/2 -0.5]
anyways you can generalize it to the matrix whose entries are like the identity matrix, but the 2x2 matrix at the end of the main diagonal, defining it as this one
floral nova
Consider the matrix A=[1 0 0; 0 -0.5 -sqrt(3)/2; 0 sqrt(3)/2 -0.5]
for n = 3, this describes what i mean
formal heath
how do you deduce that J^3?
We have I = PJ^3P^-1. Now just multiply by P and P^-1 and they cancel each other on the left
Those MSE questions have extra restriction. That the diagonalization cannot be over R but over complex numbers it is fine
The example you gave is diagonalizable over the complex numbers
floral nova
We have I = PJ^3P^-1. Now just multiply by P and P^-1 and they cancel each other...
Right lmao, i have to read better
floral nova
Those MSE questions have extra restriction. That the diagonalization cannot be o...
Assumed she referred to R, yeah
formal heath
she*
floral nova
Hm
edgy slate
This also generalizes nicely if you replace the power 3 by anything positive and...
for that you need to prove that any nontrivial jordan block raised to any natural >0 power is not an identity matrix
which is quite easy but just pointing out
grand nymph
an example would be taking a matrix associated to a rotation of 120º
Oh this is probably the idea I am looking for!