Diagonalising an n*n matrix analytically

[A Dhingra]

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!

 

[mfb]

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).

 

[A Dhingra]

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

 

[HallsofIvy]

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.

 

[Hawkeye18]

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.