Row operations¶
We revisit the previous example using algebra to express the row operations on \(A\).
A = [2 0 4 3 ; -4 5 -7 -10 ; 1 15 2 -4.5 ; -2 0 2 -13];
We use the identity and its columns heavily.
using LinearAlgebra
I4 = I(4)
4×4 Diagonal{Bool,Array{Bool,1}}:
1 ⋅ ⋅ ⋅
⋅ 1 ⋅ ⋅
⋅ ⋅ 1 ⋅
⋅ ⋅ ⋅ 1
The first step is to put a zero in the (2,1) location using a multiple of row 1:
mult21 = A[2,1]/A[1,1]
L21 = I4 - mult21*I4[:,2]*I4[:,1]'
A = L21*A
4×4 Array{Float64,2}:
2.0 0.0 4.0 3.0
0.0 5.0 1.0 -4.0
1.0 15.0 2.0 -4.5
-2.0 0.0 2.0 -13.0
We repeat the process for the (3,1) and (4,1) entries.
mult31 = A[3,1]/A[1,1];
L31 = I4 - mult31*I4[:,3]*I4[:,1]';
A = L31*A;
mult41 = A[4,1]/A[1,1];
L41 = I4 - mult41*I4[:,4]*I4[:,1]';
A = L41*A
4×4 Array{Float64,2}:
2.0 0.0 4.0 3.0
0.0 5.0 1.0 -4.0
0.0 15.0 0.0 -6.0
0.0 0.0 6.0 -10.0
And so on, following the pattern as before.