Spectrum of the Reduced matrix's eigenvalues

In summary, the conversation revolves around the question of whether the density matrix spectrum is always discrete or if it can have a continuum spectrum. While a pure density matrix has a discrete spectrum, it is not clear for general cases. The concept of compact operators having discrete eigenvalues and a possible accumulation point of zero is mentioned, with the question of whether this applies to reduced density matrices. The speaker also asks for recommendations on where to find more discussion on this topic.
  • #1
fpaolini
4
0
I would like to know if the density matrix spectrum is always discrete or if it is possible it has a continuum spectrum. It is clear that a pure density matrix has a discrete spectrum but it is not obvious in general.

I have heard that all compact operator has discrete eigenvalues and if it has an accumulation point it must be zero. It seems to me to be the case for reduced density matrix but as I am not a good mathematician I cannot see if a reduced matrix is or not a compact operator

Where could I find some discussion about that topic?
Thanks.
 
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  • #2
Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

Related to Spectrum of the Reduced matrix's eigenvalues

What is the spectrum of the reduced matrix's eigenvalues?

The spectrum of the reduced matrix's eigenvalues refers to the set of all eigenvalues of the reduced matrix. These eigenvalues represent the solutions to a system of linear equations and can provide insights into the behavior and properties of the matrix.

How is the spectrum of the reduced matrix's eigenvalues calculated?

The spectrum of the reduced matrix's eigenvalues can be calculated using various methods, such as the power method, inverse iteration, or QR algorithm. These methods involve finding the eigenvalues of the matrix iteratively until they converge to a solution.

What is the significance of the spectrum of the reduced matrix's eigenvalues?

The spectrum of the reduced matrix's eigenvalues can reveal important information about the matrix, such as its size, rank, and invertibility. It can also provide insights into the stability and behavior of a system represented by the matrix.

How does the spectrum of the reduced matrix's eigenvalues relate to the original matrix?

The spectrum of the reduced matrix's eigenvalues is closely related to the eigenvalues of the original matrix. In fact, the eigenvalues of the reduced matrix are a subset of the eigenvalues of the original matrix. However, the reduced matrix may have fewer eigenvalues if the original matrix is not invertible.

How can the spectrum of the reduced matrix's eigenvalues be used in practical applications?

The spectrum of the reduced matrix's eigenvalues has various applications in fields such as engineering, physics, and economics. It can be used to analyze the stability of systems, solve differential equations, and model complex phenomena. It is also used in data compression techniques and image processing algorithms.

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