Help with LU Decomposition and Eigenvectors

Okay, this is a two part question and it is more theoretical.

>> A=[12 -0.4 7; 6 -3.2 0.5; 2 -1 5], b=[1;1;1];

Question 1: [Q R S]= lu(A) Can someone please explain to me explicitly what Q, R, and S represent as outputs? Here are the actual outputs:

>> [Q R S]=lu(A)

Q =

      1.0000         0         0
      0.5000    1.0000         0
      0.1667    0.3111    1.0000

R =

     12.0000   -0.4000    7.0000
           0   -3.0000   -3.0000
           0         0    4.7667

S =

       1     0     0
       0     1     0
       0     0     1

Question 2: [Alpha Beta]= eig(A). Can someone please explain explicitly what Alpha and Beta represent? I know they deal with complex answers, but that's about it. Here are my actual outputs:

>> [Alpha Beta]=eig(A)

Alpha =

     -0.9218   -0.0320   -0.5944
     -0.3421    0.9914   -0.4561
     -0.1823    0.1267    0.6623

Beta =

     13.2360         0         0
           0   -3.3297         0
           0         0    3.8936

Thank you for any help!

Answer 1

[John D'Errico]

READ THE HELP. Ok, it may be proper here to tell you to take a linear algebra course if you do not understand LU decompositions or eigenvalues and eigenvectors. There are entire books written on these subjects, so do you really expect us to do better here, off the cuff?

From the help for lu:

[L,U,P] = lu(A) returns unit lower triangular matrix L, upper
    triangular matrix U, and permutation matrix P so that P*A = L*U.

So L is a lower triangular matrix. It has the property of having a unit diagonal.

U is an upper triangular matrix.

P is a permutation matrix, that essentially does pivoting to make the problem well-posed, thus avoiding zero pivots.

If you really need more than this, for example, what a matrix factorization is or why one would build it in the first place, then you REALLY need to learn some (numerical) linear algebra. Almost any basic text will cover this in great detail.