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In [1]:

    
eye(5)









    Out[1]:





5×5 Array{Float64,2}:
 1.0  0.0  0.0  0.0  0.0
 0.0  1.0  0.0  0.0  0.0
 0.0  0.0  1.0  0.0  0.0
 0.0  0.0  0.0  1.0  0.0
 0.0  0.0  0.0  0.0  1.0



In [8]:

    
# 1-indexed!!!
eye(5)[1, 1]









    Out[8]:





1.0



In [12]:

    
eye(5)[1:2, 1:2]









    Out[12]:





2×2 Array{Float64,2}:
 1.0  0.0
 0.0  1.0



In [15]:

    
eye(5)[1:3, :]









    Out[15]:





3×5 Array{Float64,2}:
 1.0  0.0  0.0  0.0  0.0
 0.0  1.0  0.0  0.0  0.0
 0.0  0.0  1.0  0.0  0.0



In [16]:

    
eye(5)[1:3, 4]









    Out[16]:





3-element Array{Float64,1}:
 0.0
 0.0
 0.0



In [22]:

    
x = repeat([1], outer=[3])









    Out[22]:





3-element Array{Int64,1}:
 1
 1
 1



In [28]:

    
*(x, transpose(x))









    Out[28]:





3×3 Array{Int64,2}:
 1  1  1
 1  1  1
 1  1  1



In [33]:

    
e = eye(4) + 3 * *([0, 0, 0, 1], transpose([1, 0, 0, 0]))
e









    Out[33]:





4×4 Array{Float64,2}:
 1.0  0.0  0.0  0.0
 0.0  1.0  0.0  0.0
 0.0  0.0  1.0  0.0
 3.0  0.0  0.0  1.0

Numerical Algorithms by Justin Solomon

https://people.csail.mit.edu/jsolomon/share/book/numerical_book.pdf

Example 3.3 Page 54



In [96]:

    
A = [[0 1 -1]; [3 -1 1]; [1 1 -2]]









    Out[96]:





3×3 Array{Int64,2}:
 0   1  -1
 3  -1   1
 1   1  -2



In [87]:

    
# permute
P = [[0 0 1]; [1 0 0]; [0 1 0]]









    Out[87]:





3×3 Array{Int64,2}:
 0  0  1
 1  0  0
 0  1  0



In [88]:

    
e3_ = repeat([0], outer=[3, 1])
e3_[3, 1] = 1
e1_T = repeat([0], outer=[1, 3])
e1_T[1, 1] = 1
E1 = eye(3) - 3 * *(e3_, e1_T)









    Out[88]:





3×3 Array{Float64,2}:
  1.0  0.0  0.0
  0.0  1.0  0.0
 -3.0  0.0  1.0



In [64]:

    
e2_T = repeat([0], outer=[1, 3])
e2_T[1, 2] = 1
E2 = eye(3) + 4 * *(e3_, e2_T)









    Out[64]:





3×3 Array{Float64,2}:
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  4.0  1.0



In [66]:

    
S = Diagonal([1,1,1/3])









    Out[66]:





3×3 Diagonal{Float64}:
 1.0   ⋅    ⋅      
  ⋅   1.0   ⋅      
  ⋅    ⋅   0.333333



In [72]:

    
e1_ = repeat([0], outer=[3, 1])
e1_[1, 1] = 1
e3_T = repeat([0], outer=[1, 3])
e3_T[1, 3] = 1
E3 = eye(3) + 2 * *(e1_, e3_T)









    Out[72]:





3×3 Array{Float64,2}:
 1.0  0.0  2.0
 0.0  1.0  0.0
 0.0  0.0  1.0



In [73]:

    
e2_ = repeat([0], outer=[3, 1])
e2_[2, 1] = 1
E4 = eye(3) + *(e2_, e3_T)









    Out[73]:





3×3 Array{Float64,2}:
 1.0  0.0  0.0
 0.0  1.0  1.0
 0.0  0.0  1.0



In [76]:

    
e3_T









    Out[76]:





1×3 Array{Int64,2}:
 0  0  1



In [77]:

    
*(e1_, e3_T)









    Out[77]:





3×3 Array{Int64,2}:
 0  0  1
 0  0  0
 0  0  0



In [89]:

    
E5 = eye(3) - *(e1_, e2_T)









    Out[89]:





3×3 Array{Float64,2}:
 1.0  -1.0  0.0
 0.0   1.0  0.0
 0.0   0.0  1.0



In [91]:

    
A_inv = E5 * E4 * E3 * S * E2 * E1 * P









    Out[91]:





3×3 Array{Float64,2}:
 0.333333  0.333333   0.0
 2.33333   0.333333  -1.0
 1.33333   0.333333  -1.0



In [102]:

    
A_inv









    Out[102]:





3×3 Array{Float64,2}:
 0.333333  0.333333   0.0
 2.33333   0.333333  -1.0
 1.33333   0.333333  -1.0



In [103]:

    
b = [-1, 4, -3]









    Out[103]:





3-element Array{Int64,1}:
 -1
  4
 -3



In [106]:

    
# Solution
A_inv * b









    Out[106]:





3-element Array{Float64,1}:
 1.0
 2.0
 3.0



In [105]:

    
# Identity
A_inv * A









    Out[105]:





3×3 Array{Float64,2}:
 1.0  -5.55112e-17  5.55112e-17
 0.0   1.0          2.22045e-16
 0.0   0.0          1.0

LU factorization



In [114]:

    
m = [[1 2 3]; [4 5 6]; [7 8 9]]









    Out[114]:





3×3 Array{Int64,2}:
 1  2  3
 4  5  6
 7  8  9



In [115]:

    
l, u, p = lu(A)









    Out[115]:





([1.0 0.0 0.0; 0.333333 1.0 0.0; 0.0 0.75 1.0], [3.0 -1.0 1.0; 0.0 1.33333 -2.33333; 0.0 0.0 0.75], [2, 3, 1])



In [116]:

    
l









    Out[116]:





3×3 Array{Float64,2}:
 1.0       0.0   0.0
 0.333333  1.0   0.0
 0.0       0.75  1.0



In [117]:

    
u









    Out[117]:





3×3 Array{Float64,2}:
 3.0  -1.0       1.0    
 0.0   1.33333  -2.33333
 0.0   0.0       0.75



In [118]:

    
p









    Out[118]:





3-element Array{Int64,1}:
 2
 3
 1