Proving tr(A A^T) = tr (A^T A)

  • Thread starter ha9981
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In summary, The conversation discusses the equality of tr(A B^T) and tr(B A^T), with the possibility that tr(A B^T) = tr(B A^T) if the trace is cyclic. The speaker mentions attempting to prove this equality in various ways and suggests considering the formula for matrix multiplication to prove the cyclic property.
  • #1
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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  • #2
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 [tex]a_{i,i}[/tex]

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)
 

1. What does "tr" stand for in the equation "tr(A A^T) = tr (A^T A)"?

"tr" stands for the trace of a matrix, which is the sum of the elements on the main diagonal of the matrix.

2. Why is it important to prove that tr(A A^T) = tr (A^T A)?

This equation is important because it shows that the trace of a matrix is invariant under transpose operations. This means that no matter the order in which we transpose a matrix and multiply it by its original, the trace will remain the same.

3. How can I prove that tr(A A^T) = tr (A^T A)?

There are a few different ways to prove this equation, but one method is to use the properties of the trace, such as linearity and the fact that the trace of a product is equal to the product of the traces. Another method is to expand the matrices and use the commutative property of multiplication.

4. Does this equation only apply to square matrices?

Yes, this equation only applies to square matrices because the trace is only defined for square matrices.

5. What is the significance of tr(A A^T) = tr (A^T A) in linear algebra?

This equation is significant in linear algebra because it is one of the properties that helps us understand and manipulate matrices. It also has practical applications in areas such as statistics and physics, where the trace is used to calculate important quantities like the variance of a matrix.

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