In [1]:

using LinearAlgebra

Square Case first¶

In [3]:

A = rand(4,4)

Out[3]:

4×4 Array{Float64,2}:
 0.330869  0.843845  0.0623888  0.398208
 0.8661    0.204488  0.138221   0.218923
 0.741009  0.775278  0.576722   0.9775  
 0.87276   0.139498  0.072938   0.983904

In [4]:

Q,R  = qr(A)

Out[4]:

LinearAlgebra.QRCompactWY{Float64,Array{Float64,2}}
Q factor:
4×4 LinearAlgebra.QRCompactWYQ{Float64,Array{Float64,2}}:
 -0.224588   0.76522    0.603058  -0.017861 
 -0.587892  -0.292547   0.130273  -0.742852 
 -0.502982   0.437253  -0.739298   0.0962124
 -0.592413  -0.371031   0.269792   0.662266 
R factor:
4×4 Array{Float64,2}:
 -1.47323  -0.782326  -0.428561  -1.29268 
  0.0       0.873139   0.232416   0.303028
  0.0       0.0       -0.351061  -0.188552
  0.0       0.0        0.0        0.575913

In [6]:

round.(Q'Q, digits=0)

Out[6]:

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
 -0.0  -0.0   0.0   1.0

In [8]:

round.(Q*Q', digits=0)

Out[8]:

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
 -0.0  -0.0   0.0   1.0

I'm happy, I see Q is orthogonal¶

In [9]:

A ≈ Q * R

Out[9]:

true

In [10]:

Q * R

Out[10]:

4×4 Array{Float64,2}:
 0.330869  0.843845  0.0623888  0.398208
 0.8661    0.204488  0.138221   0.218923
 0.741009  0.775278  0.576722   0.9775  
 0.87276   0.139498  0.072938   0.983904

In [11]:

Out[11]:

4×4 Array{Float64,2}:
 0.330869  0.843845  0.0623888  0.398208
 0.8661    0.204488  0.138221   0.218923
 0.741009  0.775278  0.576722   0.9775  
 0.87276   0.139498  0.072938   0.983904

Now a tall skinny example¶

In [12]:

A = rand(5,3)

Out[12]:

5×3 Array{Float64,2}:
 0.657045  0.214426  0.0462177
 0.268263  0.208357  0.269215 
 0.410459  0.948475  0.756601 
 0.391947  0.683485  0.28925  
 0.949405  0.999374  0.570501

In [15]:

Q,R = qr(A)
Q = Q[:,1:3] # make sure we have the first three columns

Out[15]:

5×3 Array{Float64,2}:
 -0.499955  -0.62267     0.00434913
 -0.204125  -0.0946497  -0.628421  
 -0.312324   0.686918   -0.424575  
 -0.298239   0.362414    0.645285  
 -0.722416   0.0110753   0.0917176

In [16]:

Out[16]:

5×3 Array{Float64,2}:
 -0.499955  -0.62267     0.00434913
 -0.204125  -0.0946497  -0.628421  
 -0.312324   0.686918   -0.424575  
 -0.298239   0.362414    0.645285  
 -0.722416   0.0110753   0.0917176

In [17]:

Out[17]:

3×3 Array{Float64,2}:
 -1.31421  -1.37177  -0.81277 
  0.0       0.75706   0.57661 
  0.0       0.0      -0.251239

In [18]:

Q * R

Out[18]:

5×3 Array{Float64,2}:
 0.657045  0.214426  0.0462177
 0.268263  0.208357  0.269215 
 0.410459  0.948475  0.756601 
 0.391947  0.683485  0.28925  
 0.949405  0.999374  0.570501

In [19]:

Out[19]:

5×3 Array{Float64,2}:
 0.657045  0.214426  0.0462177
 0.268263  0.208357  0.269215 
 0.410459  0.948475  0.756601 
 0.391947  0.683485  0.28925  
 0.949405  0.999374  0.570501

In [20]:

A ≈ Q * R

Out[20]:

true

In [22]:

round.(Q'Q, digits=0)

Out[22]:

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

In [23]:

Q * Q'

Out[23]:

5×5 Array{Float64,2}:
  0.637693    0.158256   -0.273422   -0.0737523  0.354678 
  0.158256    0.445538    0.265548   -0.378935   0.0887777
 -0.273422    0.265548    0.749666    0.0681243  0.194295 
 -0.0737523  -0.378935    0.0681243   0.636683   0.27865  
  0.354678    0.0887777   0.194295    0.27865    0.53042

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