Is tr(A A^T) Equal to tr(A^T A)?

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SUMMARY

The discussion confirms that tr(A A^T) is indeed equal to tr(A^T A) based on the cyclic property of the trace function. The proof hinges on the definition of the trace as the sum of the diagonal elements of a matrix and the properties of matrix multiplication. Specifically, if tr(AB) = tr(BA), it follows that tr(AA^T) = tr(A^T A). This conclusion is established through a straightforward application of the cyclic property of the trace.

PREREQUISITES
  • Understanding of matrix multiplication
  • Familiarity with the concept of the trace of a matrix
  • Knowledge of linear algebra properties, specifically the cyclic property of the trace
  • Basic skills in mathematical proof techniques
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  • Study the properties of matrix traces in linear algebra
  • Learn about the cyclic property of the trace in detail
  • Explore examples of matrix multiplication and their implications on trace
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Students and professionals in mathematics, particularly those studying linear algebra, as well as researchers and educators looking to deepen their understanding of matrix properties and proofs.

ha9981
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to begin I am wondering if its even true that they are equal. As i lost the sheet with that on it. If that is not true it could have been tr(A B^T) = tr(B A^T) but it doubt it.

I have tried proving it in so many ways, but I am stuck.
 
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well so the proof is easy if you know that the trace is cyclic (then it is just one line actually...)

assuming that you are not allowed to use this property, or at least must prove it first...

to prove that it is cyclical, notice that the trace of a matrix A is the sum a_{i,i}

try writing out the formula for matrix multiplication of two arbitrary matrices A and B (ie, what is the i,jth element of the product AB?) and then think about the case i=j

if tr(AB) = tr(BA), then the trace is cyclic... and then tr(AA^T) = tr(A^TA)
 

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