We begin with a symmetric .
A = [ 2 4 4 2
4 5 8 -5
4 8 6 2
2 -5 2 -26 ];
Carrying out our usual elimination in the first column leads us to
L1 = eye(4); L1(2:4,1) = [-2;-2;-1];
A1 = L1*A
A1 =
2 4 4 2 0 -3 0 -9 0 0 -2 -2 0 -9 -2 -28
But now let's note that if we transpose this result, we have the same first column as before! So we could apply again and then transpose back.
A2 = (L1*A1')'
A2 =
2 0 0 0 0 -3 0 -9 0 0 -2 -2 0 -9 -2 -28
Using transpose identities, this is just
A2 = A1*L1'
A2 =
2 0 0 0 0 -3 0 -9 0 0 -2 -2 0 -9 -2 -28
Now you can see how we proceed down and to the right, eliminating in a column and then symmetrically in the corresponding row.
L2 = eye(4); L2(3:4,2) = [0;-3];
A3 = L2*A2*L2'
A3 =
2 0 0 0 0 -3 0 0 0 0 -2 -2 0 0 -2 -1
Finally, we arrive at a diagonal matrix.
L3 = eye(4); L3(4,3) = -1;
D = L3*A3*L3'
D =
2 0 0 0 0 -3 0 0 0 0 -2 0 0 0 0 1