Linear Dependence of Matrix Vectors

zplot
Messages
17
Reaction score
0
I need to prove that the set {I, A, A^2,..., A^n} is linear dependent where A is any nxn matrix. The vector space is the set of nxn matrix, considered as a nxn dimensional vector space.

Does anybody have an idea how to prove it?
Thank you very much.
 
Physics news on Phys.org
well I maybe wrong but to get you started:

I is in the nxn vector space and all the other matrices A, A^2, ..A^n

You might start by showing the linear combinations of matrices to get I and then relate that to the matrices A, of which matrices' can be written as nxn A^n=PA(^n)P^(-1)
 
Well, I tried to write A=P^(-1) J P where J is a Jordan matrix. Even more, I also tried to put J=N + D where N is nilpotent such that N^n=0 and D diagonal but I could not prove that the set is linear dependent. Thank you for your help. I f you have any further details or ideas I would be pleased.
 
At the end I arrived at the right solution. (A-lambda I)^k must be zero for some k<n. It comes from the Jordan canonical matrix, where lambda belongs to its spectrum. Logically then, the set {I, A, A^2,..., A^n} is linear dependent.

Thank you
 
An alternative method, which gives you the exact dependence right off the bat, is to use the Cayley-Hamilton theorem. The characteristic polynomial is of degree at most n, and is satisfied by the matrix.
 
Great! Thats certainly a much better, simpler and more elegant solution.
Thank you very much, Henry.
 
Thread 'Derivation of equations of stress tensor transformation'
Hello ! I derived equations of stress tensor 2D transformation. Some details: I have plane ABCD in two cases (see top on the pic) and I know tensor components for case 1 only. Only plane ABCD rotate in two cases (top of the picture) but not coordinate system. Coordinate system rotates only on the bottom of picture. I want to obtain expression that connects tensor for case 1 and tensor for case 2. My attempt: Are these equations correct? Is there more easier expression for stress tensor...
Back
Top