In [15]:

using LinearAlgebra

In [16]:

function fullsvd(A) 
    U,s,V = svd(A, full = true)  # compute svd
    Σ = zeros(size(A))           # container for Σ  
    for i=1:length(s)
       Σ[i,i] = s[i]            # place singular values in Σ
    end                      # a practical svd would never store all these zeros
    display(U);display(Σ);display(V) # display the answer
    return(U,Σ,V)                # return the answer
end


function rankrsvd(A) 
    U,s,V = svd(A, full = true)  # compute svd
    r = sum(s.>1e-8)             # rank = how many positive?
    U₁ = U[:,1:r]
    Σᵣ = Diagonal(s[1:r])        # Diagonal matrix of singular values
    V₁ = V[:,1:r]
    display(U₁);display(Σᵣ);display(V₁) # display the answer
    return(U₁,Σᵣ,V₁)                # return the answer
end

Out[16]:

rankrsvd (generic function with 1 method)

1. random 2x2 matrix¶

In [17]:

A = rand(2,2)

Out[17]:

2×2 Array{Float64,2}:
 0.259439  0.075927
 0.898109  0.918728

In [18]:

fullsvd(A);

2×2 Array{Float64,2}:
 -0.18222   -0.983258
 -0.983258   0.18222

2×2 Array{Float64,2}:
 1.30643  0.0    
 0.0      0.13025

2×2 Adjoint{Float64,Array{Float64,2}}:
 -0.712128  -0.70205 
 -0.70205    0.712128

In [19]:

rankrsvd(A);

2×2 Array{Float64,2}:
 -0.18222   -0.983258
 -0.983258   0.18222

2×2 Diagonal{Float64,Array{Float64,1}}:
 1.30643   ⋅     
  ⋅       0.13025

2×2 Array{Float64,2}:
 -0.712128  -0.70205 
 -0.70205    0.712128

2. random 3x2 matrix¶

In [20]:

A = rand(3,2)

Out[20]:

3×2 Array{Float64,2}:
 0.464581  0.051883
 0.9702    0.533329
 0.601868  0.413574

In [21]:

fullsvd(A);

3×3 Array{Float64,2}:
 -0.311792   0.90137     0.300529
 -0.794641  -0.0739732  -0.602555
 -0.520894  -0.426685    0.73933

3×2 Array{Float64,2}:
 1.39313  0.0     
 0.0      0.191691
 0.0      0.0

2×2 Adjoint{Float64,Array{Float64,2}}:
 -0.882421   0.47046 
 -0.47046   -0.882421

In [22]:

rankrsvd(A);

3×2 Array{Float64,2}:
 -0.311792   0.90137  
 -0.794641  -0.0739732
 -0.520894  -0.426685

2×2 Diagonal{Float64,Array{Float64,1}}:
 1.39313   ⋅      
  ⋅       0.191691

2×2 Array{Float64,2}:
 -0.882421   0.47046 
 -0.47046   -0.882421

3. random 2x3 matrix¶

In [23]:

A = rand(2,3)

Out[23]:

2×3 Array{Float64,2}:
 0.0451975  0.242917  0.405185
 0.477637   0.8663    0.725397

In [24]:

fullsvd(A);

2×2 Array{Float64,2}:
 -0.337277  -0.941405
 -0.941405   0.337277

2×3 Array{Float64,2}:
 1.30125  0.0       0.0
 0.0      0.191821  0.0

3×3 Adjoint{Float64,Array{Float64,2}}:
 -0.357268   0.618008  -0.700304
 -0.689699   0.331038   0.643994
 -0.629821  -0.713078  -0.30797

In [25]:

rankrsvd(A);

2×2 Array{Float64,2}:
 -0.337277  -0.941405
 -0.941405   0.337277

2×2 Diagonal{Float64,Array{Float64,1}}:
 1.30125   ⋅      
  ⋅       0.191821

3×2 Array{Float64,2}:
 -0.357268   0.618008
 -0.689699   0.331038
 -0.629821  -0.713078

4. rank 3, 7x10 matrix¶

In [26]:

A = rand(7,3)*rand(3,10) # this should be rank 3

Out[26]:

7×10 Array{Float64,2}:
 0.409585  0.550866  0.729917  0.396048  …  0.491844  0.870667  0.327258
 0.824315  0.457974  0.887333  0.932895     0.34797   1.19361   0.266938
 0.897034  0.338666  0.678985  0.943025     0.418739  1.0996    0.258406
 0.862495  0.661816  0.926009  0.816466     0.771865  1.33408   0.463019
 0.878864  0.887087  1.42046   0.99785      0.602456  1.66596   0.472172
 0.765583  0.423397  0.731421  0.800129  …  0.446203  1.06758   0.287248
 0.710849  0.713791  1.12556   0.793167     0.511149  1.33563   0.388595

In [27]:

fullsvd(A);

7×7 Array{Float64,2}:
 -0.277937  -0.522311   -0.0744039  …   0.466151  -0.560653    0.264395   
 -0.34423    0.518096   -0.0720404     -0.227442  -0.460716   -0.319634   
 -0.335511   0.428918    0.464644       0.604044   0.0486161  -0.0705107  
 -0.436358  -0.497343    0.506287      -0.350516   0.117976   -0.410459   
 -0.487624   0.0167782  -0.553854      -0.204899  -0.105282   -0.000703896
 -0.327734   0.157075    0.264594   …  -0.34661    0.116364    0.806582   
 -0.395088  -0.0499189  -0.374552       0.284817   0.657649   -0.0625039

7×10 Array{Float64,2}:
 6.92226  0.0       0.0       0.0          …  0.0          0.0  0.0  0.0
 0.0      0.755198  0.0       0.0             0.0          0.0  0.0  0.0
 0.0      0.0       0.683053  0.0             0.0          0.0  0.0  0.0
 0.0      0.0       0.0       5.91088e-16     0.0          0.0  0.0  0.0
 0.0      0.0       0.0       0.0             0.0          0.0  0.0  0.0
 0.0      0.0       0.0       0.0          …  0.0          0.0  0.0  0.0
 0.0      0.0       0.0       0.0             8.21229e-17  0.0  0.0  0.0

10×10 Adjoint{Float64,Array{Float64,2}}:
 -0.294011   0.35548     0.312083   …   0.0465244    0.0514862   0.039223 
 -0.2263    -0.249712   -0.334077       0.181558     0.595424    0.0233641
 -0.363646  -0.0109916  -0.510499      -0.0359443   -0.220601    0.140677 
 -0.312911   0.500153    0.171036      -0.359502    -0.207792    0.0736223
 -0.237955  -0.64866     0.255552      -0.601864    -0.131402   -0.104849 
 -0.251608   0.0397458  -0.437401   …   0.164673    -0.485916   -0.221939 
 -0.477004  -0.0809271   0.436915       0.478804     0.100636    0.0399163
 -0.198741  -0.299539    0.17074        0.373694    -0.353637   -0.236875 
 -0.475836   0.133415   -0.153594      -0.275708     0.389516   -0.22132  
 -0.137166  -0.156822   -0.0295032      0.00811763  -0.103379    0.897647

In [28]:

rankrsvd(A);

7×3 Array{Float64,2}:
 -0.277937  -0.522311   -0.0744039
 -0.34423    0.518096   -0.0720404
 -0.335511   0.428918    0.464644 
 -0.436358  -0.497343    0.506287 
 -0.487624   0.0167782  -0.553854 
 -0.327734   0.157075    0.264594 
 -0.395088  -0.0499189  -0.374552

3×3 Diagonal{Float64,Array{Float64,1}}:
 6.92226   ⋅         ⋅      
  ⋅       0.755198   ⋅      
  ⋅        ⋅        0.683053

10×3 Array{Float64,2}:
 -0.294011   0.35548     0.312083 
 -0.2263    -0.249712   -0.334077 
 -0.363646  -0.0109916  -0.510499 
 -0.312911   0.500153    0.171036 
 -0.237955  -0.64866     0.255552 
 -0.251608   0.0397458  -0.437401 
 -0.477004  -0.0809271   0.436915 
 -0.198741  -0.299539    0.17074  
 -0.475836   0.133415   -0.153594 
 -0.137166  -0.156822   -0.0295032

In [ ]: