Determinat of a continous matrix?

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In summary, the conversation discusses the calculation of the determinant or inverse matrix of a continuous matrix using real numbers as row and column indices. An example is given with a Hamiltonian in momentum representation, which is an infinite dimensional matrix. It is mentioned that if the matrix elements are real, the matrix is symmetric and can be diagonalized using an orthogonal matrix. The determinant can then be calculated using the trace and natural logarithm of the diagonalized matrix. The conversation ends with a question on how to find the eigen vectors if only the matrix elements are known.
  • #1
wenty
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Determinat of a "continous" matrix?

a(x,y) is a matrix element ,and x,y is the row and column index.
If x,y are real numbers, how to calculate the determinat of this matrix or the inverse matrix?

An example of this kind of matrix is <k|H|k'>,a Hamiltonian in momentum representation.
 
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  • #2
So it's an infinite dimensional matrix...

If the matrix elements [itex] \hat{H}_{k,k'}=:\langle k|\hat{H}|k' \rangle [/itex] are real,then this matrix is symmetric.Then u can find an orthogonal matrix which would diagonalize the hamiltonian matrix.

Then

[tex] \det\left(\hat{H}_{k,k'}}\right)=exp \ \left(trace \ ln \hat{H}^{diag}_{k,k'}\right) [/tex]



Daniel.
 
  • #3
dextercioby said:
So it's an infinite dimensional matrix...

If the matrix elements [itex] \hat{H}_{k,k'}=:\langle k|\hat{H}|k' \rangle [/itex] are real,then this matrix is symmetric.Then u can find an orthogonal matrix which would diagonalize the hamiltonian matrix.

Then

[tex] \det\left(\hat{H}_{k,k'}}\right)=exp \ \left(trace \ ln \hat{H}^{diag}_{k,k'}\right) [/tex]



Daniel.

The problem is how to find this orthogonal matrix.In ordinary case,to find this matrix we should solve an equation which needs to know the determinant of the given matrix.Then the question remains.
 
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  • #4
How about u search for a basis (a new set of kets [itex] |k\rangle [/itex]) made up of eigen vectors (in general sense,the spectrum in continuous) of the Hamiltonian,and then everything would be tremendously simple...?

Daniel.
 
  • #5
dextercioby said:
How about u search for a basis (a new set of kets [itex] |k\rangle [/itex]) made up of eigen vectors (in general sense,the spectrum in continuous) of the Hamiltonian,and then everything would be tremendously simple...?

Daniel.



But if what you only know is a(i,j),which isn's the matrix elements of the Hamiltonian,how do you find the eigen vectors?
 

1. What is a continuous matrix?

A continuous matrix is a mathematical representation of a set of continuous data. It is a rectangular array of numbers that can be used to analyze and manipulate continuous functions.

2. How is the determinant of a continuous matrix calculated?

The determinant of a continuous matrix is calculated by using a mathematical formula that involves the elements of the matrix. The formula is different for different sized matrices, but it ultimately involves multiplying and adding the elements in a specific pattern.

3. What is the significance of the determinant of a continuous matrix?

The determinant of a continuous matrix is a measure of how the matrix transforms space. It can be used to determine whether a matrix has an inverse, and it also has applications in solving systems of linear equations and finding eigenvalues and eigenvectors.

4. Can the determinant of a continuous matrix be negative?

Yes, the determinant of a continuous matrix can be negative. The sign of the determinant depends on the arrangement of the elements in the matrix, and it can be positive, negative, or zero.

5. How is the determinant of a continuous matrix used in real-world applications?

The determinant of a continuous matrix has many real-world applications, such as in computer graphics, physics, economics, and engineering. It is used to analyze and manipulate data in a variety of fields, and it can provide valuable insights and solutions to complex problems.

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