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Statistical mechanics problem

  • Thread starter Send BoBs
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  • #1
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Homework Statement:

Prove that Tr(αA+βB) = αTr(A)+βTr(B). α and β are complex constants, and A and B are dXd matrices

Homework Equations:

Tr(αA+βB) = αTr(A)+βTr(B)
First, i'd like to apologize for the vague title. Unfortunately my understanding of the question is equally vague. I think the dXd matrix is meant to be a covariance matrix, so the above equation would be some complex constant multiplied by the covariance matrix. The Tr would referring to the trace of the matrix or sum of diagonal elements. So i'm attempting to show that the "trace of the sum A+B" is equal to "trace A + trace B".

Here's my main problem. I have never heard of a covarience matrix before. If someone could show me a simple example of what a covarience matrix is then I should be able to figure out the additive, multiplicative, etc... rules of these matrices.
 

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  • #2
TeethWhitener
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Homework Statement: Prove that Tr(αA+βB) = αTr(A)+βTr(B). α and β are complex constants, and A and B are dXd matrices
Homework Equations: Tr(αA+βB) = αTr(A)+βTr(B)

I think the dXd matrix is meant to be a covariance matrix,
The statement to be proven is true for all square matrices, not just covariance matrices. Try writing a general expression for the trace of a matrix.
 
  • #3
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The statement to be proven is true for all square matrices, not just covariance matrices. Try writing a general expression for the trace of a matrix.
Thank you. Clearly i'm just getting confused by new terms and not giving this a proper thought. I should probably take some time to get more familiar with the notation used for statistical mechanics.
 

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WWGD
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Send Bob's:Trace is just the sum of diagonal entries. Can you take it from there?
 
  • #5
Delta2
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This is rather a linear algebra problem not a statistical mechanics one, and I think the proof is 2-3 lines max.
The matrix ##C=\alpha A+\beta B## has as diagonal elements ##c_{ii}=\alpha a_{ii}+\beta b_{ii}## where ##a_{ii},b_{ii}## are the diagonal elements of the matrices A and B respectively.
What is the trace of ##C##, ##Tr(C)## with respect to the diagonal elements ##c_{ii}##? Proceed from here and using simple properties of a finite sum you should be able to prove the result.
 

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