Do Commuting Linear Operators A and B Satisfy the Exponential Property?

AI Thread Summary
The discussion centers on the relationship between two commuting linear operators A and B, specifically regarding the exponential property e^A * e^B = e^(A+B). This property holds true if A and B commute, meaning AB = BA. The question also explores whether the property is valid when applied to an eigenvector Y of both operators, which is always true. Additionally, it is noted that A and B must be of the same size and square matrices for eigenvectors and values to exist. The necessity of the commutation condition for the exponential property remains uncertain.
frederick
Messages
2
Reaction score
0
If A & B are linear operators, and AY=aY & BY=bY, what is the relationship between A & B such that e^A*e^B=e^(A+B)?? --where e^x=1+x+x^2/2+x^3/3!+...+x^n/n!
 
Last edited:
Mathematics news on Phys.org
e^A e^B = e^(A+B) is true when A and B commute (AB=BA). Or are you asking when e^A e^B Y=e^(A+B) Y, where Y is an eigenvector of A and B? That is always true.
 
Last edited:
e^A*e^B=e^(A+B)??

A and B commute is sufficient - I don't know if it's necessary.
 
I'm not quite sure what the question is asking here...

I definitely did pick up that A and B must be the same size for A+B to exist (and must be square to have eigenvectors and values). Also, you need AB=BA (I think?). Someone else should verify this
 
Thread 'Video on imaginary numbers and some queries'
Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find the difference between two positive quantities, but you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Thread 'Unit Circle Double Angle Derivations'
Here I made a terrible mistake of assuming this to be an equilateral triangle and set 2sinx=1 => x=pi/6. Although this did derive the double angle formulas it also led into a terrible mess trying to find all the combinations of sides. I must have been tired and just assumed 6x=180 and 2sinx=1. By that time, I was so mindset that I nearly scolded a person for even saying 90-x. I wonder if this is a case of biased observation that seeks to dis credit me like Jesus of Nazareth since in reality...
Back
Top