Is the Trace Method the Key to Proving Ab-ba=i Has No Solution?

  • Context: Undergrad 
  • Thread starter Thread starter GreenApple
  • Start date Start date
Click For Summary
SUMMARY

The discussion centers on the mathematical proof that there are no real square matrices A and B such that AB - BA = I, where I is the identity matrix. The key to this proof is the concept of the trace of a matrix. Participants highlight that using the trace method effectively demonstrates the impossibility of such matrices existing. Additionally, there is curiosity about the origins of the trace concept, suggesting it may have been developed to address similar problems.

PREREQUISITES
  • Understanding of matrix operations, specifically multiplication and subtraction.
  • Familiarity with the concept of the identity matrix.
  • Knowledge of the trace of a matrix and its properties.
  • Basic linear algebra concepts, including square matrices.
NEXT STEPS
  • Study the properties of the trace of matrices in detail.
  • Explore proofs related to the commutation of matrices and their implications.
  • Learn about the implications of the identity matrix in linear transformations.
  • Investigate other mathematical methods for proving matrix equations.
USEFUL FOR

Mathematics students, particularly those studying linear algebra, educators teaching matrix theory, and researchers interested in matrix properties and proofs.

GreenApple
Messages
30
Reaction score
0
Hi,I am a Chinese sophomore major in software engineering.I am reading Artin's Algebrarecently and have come across this problem in 1.1,and have been trying for 4 days in vain:cry:

Give me some real thought guys,I will really appreciate it!
 
Physics news on Phys.org
Sorry,the problem should be this:prove AB-BA=I has no solution with realnumber where A and B are matrix
 
I suppose you mean that you want to prove that there are no real square matrices A and B such that AB - BA = I. The key to proving this: Trace.
 
It's probably way out of my area, but let's say you take a 1x1 matrix for A and a 1x1 matrix for B. Then A*B will equal B*A...because there are no additions or substractions etc inside the matrixes...and since they're both 1*1 they can be multiplied...wouldn't matrix I end up being just ?

[0]

I probably said something very stupid...but it seems to me that it follows all the question parts...it's a matrix, it's real, it's a solution...and A and B can be anything...
 
Real matrices are those matrices which have real entries. They are not necessarily 1x1.
 
Oh so you have to prove the identity for all of them? I think what I was trying to do is find one case that works. Yeah...I'm way over my head...:(
 
thanks!

Yeah,using trace works!
Another question:is trace invented just to prove this problem?I believe that a new concept usually come from a new mothod proving something.
 

Similar threads

Matrices: if AB=A and BA=B, then B^2 is equal to?
  • · Replies 18 ·
Replies
18
Views
4K
Graduate Is Rank(A) Equal to Trace(AA*) and When Does AB-BA=I Hold?
  • · Replies 3 ·
Replies
3
Views
4K
Graduate Second derivative of a complex matrix
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Understanding Lorentz Groups and some key subgroups
  • · Replies 3 ·
Replies
3
Views
2K
Prove trace of matrix: Tr(AB) = Tr(BA)
  • · Replies 5 ·
Replies
5
Views
12K
Preparing for Physics Grad School after Math/Comp Sci undergrad
  • · Replies 8 ·
Replies
8
Views
3K
I am Looking for an Astronomy undergrad program at 50
  • · Replies 1 ·
Replies
1
Views
1K
Courses Taking Grad Courses without getting a BA in Physics?
  • · Replies 3 ·
Replies
3
Views
2K
BSc Physics: University Options for a second bachelor's degree
  • · Replies 20 ·
Replies
20
Views
4K
Graduate Proving Integer Solutions for x^3 + y^3 = z^3
  • · Replies 1 ·
Replies
1
Views
4K