What is the Connection Between Matrix Trace and Endomorphism?

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In summary, the conversation discusses the possibility of an endomorphism f on ℝ4 with bases 'a' and 'b' such that A = a[f]a and B = b[f]b. It is noted that the traces of the two matrices must be the same for this to be possible, as the trace of a linear transformation is independent of the choice of basis. It is also mentioned that conjugating a matrix by another matrix does not change the trace.
  • #1
tamintl
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Consider the 4x4 matrices
A =
(1 2 3 4)
(5 6 7 8)
(9 10 11 12)
(13 14 15 16)B=
(1 2 3 4)
(8 5 6 7)
(11 12 9 10)
(14 15 16 13)

The question I was asked was the following: Show that there does not exist an endomorphism f: ℝ4 -> ℝ4 and basis 'a' and 'b' of R^4, such that A = a[f]a and B=b[f]b.

I have read in my notes and found that if the traces of the two matrices are not the same then they cannot represent the same endomorphism.

I am struggling to see the intuition behind this though.

Can anyone shed some light?

Many thanks
 
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  • #2
The trace of a linear transformation is the sum of the eigenvalues of the matrix, and so is independent of the choice of basis.

Alternatively, you can use the fact that Tr(AB) = Tr(BA) to show that if you conjugate a matrix by another matrix the trace is unchanged.
 

1. What is trace in mathematics?

Trace is a mathematical operation that determines the sum of the elements on the main diagonal of a square matrix. It is denoted by tr(A) or Tr(A) and is an important concept in linear algebra.

2. What is an endomorphism?

An endomorphism is a linear transformation of a vector space into itself. This means that the domain and codomain of the transformation are the same. It is also known as an endomorphism operator and is commonly used in algebra and geometry.

3. How is trace related to endomorphisms?

The trace of an endomorphism is the trace of the matrix representation of the endomorphism. This means that the trace of an endomorphism is the sum of the eigenvalues of the transformation. It is also equal to the dimension of the vector space on which the endomorphism is defined.

4. What is the significance of trace and endomorphism in physics?

In physics, trace and endomorphisms are used in the study of quantum mechanics. The trace of an endomorphism is related to the expectation value of an observable in quantum mechanics. It is also used in the study of quantum entanglement and quantum information theory.

5. What are some applications of trace and endomorphism in real-world problems?

Trace and endomorphism have applications in various fields such as engineering, computer science, and economics. They are used in signal processing, image and video compression, and data analysis. In economics, they are used in game theory and optimization problems. They also have applications in cryptography and error correction codes.

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