Find eigenvalues without the function eig

Hi ! I have to write a program of the QR method for finding eigenvalues of 5x5 matrices, and I have to do that without shift and with shift calculated on the basis of an eigenvalue of the 2x2 right lower corner matrix.

And of course, I can't use the function qr or eig! So I'm a little lost and I really don't know how I can do that, if someone is able to help me, I would be really grateful. Thanks by advance !

[Nick Counts]

Hi, Mathias,

Do you:

A.) Understand the mathematics of QR decomposition and you are having trouble implementing them in Matlab?

Or,

B.) Need help with the mathematics?

Let us know where you are with this problem, what you have tried - post any code you have started and we will see what we can do to help :)

Good luck!

Answer 1

[Roger Stafford]

The eigenvector/eigenvalue problem for a square matrix A tries to solve the problem:

   (A-s)*v = 0

where ’s’ is an appropriate scalar and where ‘v’ is an eigenvector. For that reason, for a 4 by 4 matrix for example, ’s’ must satisfy the equation

   det([A11-s, A12 , A13 , A14 ;
         A21 ,A22-s, A23 , A24 ;
         A31 , A32 ,A33-s, A34 ;
         A41 , A42 , A43 ,A44-s]) = 0

The set of ’s’ values that satisfy this equation will be the eigenvalues of A.

Therefore you can use the ‘expand’ and ‘collect’ abilities of the symbolic toolbox to express this as a polynomial equation which can then be solved using the ‘roots’ function. This gives you the eigenvalues without the necessity of finding the eigenvectors.