Diagonalising an n*n matrix analytically

  • Thread starter A Dhingra
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In summary, the conversation is about diagonalizing a (2n+1)x(2n+1) matrix with specific diagonal terms and non-vanishing terms. The person is looking for a way to solve it for general n without using numerical methods. They mention a professor who used tricks to diagonalize a similar matrix but they cannot remember how. Another person suggests using the Gauß algorithm to find an analytic solution. Later, it is mentioned that there is no algebraic method for exactly solving a general polynomial of degree 5 or higher. Finally, there is a discussion about the spectrum of the matrix and references to Jacobi matrices and orthogonal polynomials.
  • #1
A Dhingra
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1
Hi everyone
I am trying to diagonalise a (2n+1)x(2n+1) matrix which has diagonal terms A_ll = (-n+l)^2 and other non vanishing terms are A_l(l+1) = A_(l+1)l = constant.
Is there any way I can solve it for general n without having to use any numerical methods.
I remember once a professor diagonalised such a matrix for a fixed value of n using some tricks, but I can't remember how he did that. Can anyone help me out here?

Any help is appreciated. Cheers!
 
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  • #2
The Gauß algorithm should need about 4n steps (+- a few), and you can do it analytically and see if the result has some reasonable expression (but even if it does not, you get an analytic result).
 
  • #3
Hi ...
I have managed to find an iterative expression to solve for the eigenvalues. But without selecting a fixed value of n I can't do anything with it. Any suggestions how I can go ahead with it.
Thanks mfb
 
  • #4
Finding the eigenvalues for a matrix is equivalent to solving the n degree eigenvalue equation. There is NO algebraic method for exactly solving a general polynomial of degree 5 or higher.
 
  • #5
Are you assuming that ##A_{j,j+1}= A_{j+1,j} = a ## for all ##j##? (i.e.> that ##a## is the same for all ##j##). If so, then the case when ##a=1## and all other entries are ##0## is the case of the so-called free Jacobi matrix. Its spectrum is computed in terms of Chebyshev polynomials. Your case then can be obtained by a simple affine transformation.

Even if your ##a##s are different, look up Jacobi matrices and orthogonal polynomials.
 

1. What is diagonalisation of a matrix?

Diagonalisation of a matrix is a process in which a square matrix is transformed into a diagonal matrix through a series of mathematical operations. This process is useful for simplifying calculations and solving systems of linear equations.

2. How is a matrix diagonalised analytically?

A matrix can be diagonalised analytically by finding its eigenvalues and eigenvectors. The eigenvalues are the values that make the determinant of the matrix equal to zero, and the eigenvectors are the corresponding vectors that satisfy the equation Ax = λx, where A is the matrix and λ is the eigenvalue. These eigenvectors are then used to construct a diagonal matrix using the formula D = P-1AP, where P is a matrix whose columns are the eigenvectors.

3. What is the importance of diagonalisation in linear algebra?

Diagonalisation is important in linear algebra because it allows for easier manipulation and analysis of matrices. Diagonal matrices have many useful properties, such as being easy to invert and being closed under multiplication, which make them useful in solving systems of linear equations and performing other calculations.

4. Can all matrices be diagonalised?

No, not all matrices can be diagonalised. A matrix can only be diagonalised if it has n linearly independent eigenvectors, where n is the size of the matrix. If a matrix does not have enough linearly independent eigenvectors, it cannot be diagonalised.

5. What are the applications of diagonalisation in real-world problems?

Diagonalisation has many applications in various fields, such as physics, engineering, and economics. It is used to solve systems of differential equations, model dynamical systems, and analyze data in statistical methods. It is also used in quantum mechanics to find the energy levels of a system.

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