How can I write the determinant as traces in this paper on complex matrices?

In summary, the conversation discusses the process of finding the logarithm of a matrix and the use of traces to simplify determinants in complex matrices. The speaker is seeking help and is advised to post their question on a math forum for quicker responses.
  • #1
rbwang1225
118
0
I am reading the paper http://arxiv.org/abs/hep-th/9701037.
In equation (2), the author write the determinat as traces, but I don't know how to do this.
I know that ##det(e^A)=e^{tr(A)}##, where ##A## is a complex matrix, and ##det(A)=e^{tr(L)}##, where ##e^L=A## and ##L## is also a complex matrix.
The problem becomes how to find the logrithm of the matrix ##g_{mn}+i##[itex]\tilde{F}_{mn}[/itex].
Above is what I can figure out now.

Any help would be vary appreciated!
Regards.
 
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  • #2
You can always post any question you have on math stackexchange .You will get responses more quickly
 

1. How do you write a determinant as a trace?

To write a determinant as a trace, you first need to understand that a trace is the sum of the diagonal elements in a matrix. The determinant of a 2x2 matrix can be written as the trace of the matrix raised to the power of 2. Similarly, the determinant of a 3x3 matrix can be written as the trace of the matrix raised to the power of 3. In general, the determinant of an nxn matrix can be written as the trace of the matrix raised to the power of n.

2. Why is it useful to write a determinant as a trace?

Writing a determinant as a trace can be useful in simplifying complicated determinants. It allows us to express the determinant in terms of the diagonal elements of the matrix, which are often easier to calculate. This can be especially useful when dealing with larger matrices.

3. Can any determinant be written as a trace?

No, not all determinants can be written as a trace. This method only works for square matrices, and even then, only for certain sizes. For example, a 4x4 matrix cannot be written as the trace of the matrix raised to the power of 4.

4. Are there other ways to write a determinant besides using traces?

Yes, there are other methods for writing determinants, such as using cofactors or expanding along a row or column. These methods may be more suitable for certain types of matrices, so it is important to understand all the different techniques for calculating determinants.

5. How can writing a determinant as a trace be applied in real-world situations?

Writing a determinant as a trace can be applied in various fields of science and engineering, such as physics, economics, and computer science. It can be used to solve systems of linear equations, calculate eigenvalues and eigenvectors, and determine the stability of dynamical systems. It is a useful tool for simplifying complex calculations and gaining a better understanding of matrix operations.

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