How to Perform QR Decomposition in R - Step by Step Examples

How to Perform QR Decomposition in R ?

Answer

To perform QR decomposition in R, you can use the qr() function. This function computes the QR decomposition of a matrix, representing it as the product of an orthogonal matrix Q and an upper triangular matrix R.


✐ Examples

1 Performing QR Decomposition on a 3x3 Matrix

In this example,

  1. We start by creating a 3x3 matrix named mat using the matrix() function. This matrix represents the data we want to decompose.
  2. Next, we use the qr() function to perform QR decomposition on the matrix mat. We assign the result to a variable named qr_res.
  3. We extract the orthogonal matrix Q from qr_res using the $qr attribute and assign it to a variable named Q.
  4. We also extract the upper triangular matrix R from qr_res using the $qr attribute and assign it to a variable named R.
  5. We print both matrices Q and R to the console to see the results. This allows us to verify the decomposition.

R Program

mat <- matrix(c(1, 2, 3, 4, 5, 6, 7, 8, 9), nrow = 3, byrow = TRUE)
qr_res <- qr(mat)
Q <- qr.Q(qr_res)
R <- qr.R(qr_res)
print('Orthogonal Matrix Q:')
print(Q)
print('Upper Triangular Matrix R:')
print(R)

Output

[1] "Orthogonal Matrix Q:"
           [,1]       [,2]       [,3]
[1,] -0.1230915  0.9045340  0.4082483
[2,] -0.4923660  0.3015113 -0.8164966
[3,] -0.8616404 -0.3015113  0.4082483
[1] "Upper Triangular Matrix R:"
          [,1]      [,2]          [,3]
[1,] -8.124038 -9.601136 -1.107823e+01
[2,]  0.000000  0.904534  1.809068e+00
[3,]  0.000000  0.000000 -2.220446e-16

2 Performing QR Decomposition on a 4x2 Matrix

In this example,

  1. We start by creating a 4x2 matrix named mat using the matrix() function. This matrix represents another set of data we want to decompose.
  2. Next, we use the qr() function to perform QR decomposition on the matrix mat. We assign the result to a variable named qr_res.
  3. We extract the orthogonal matrix Q from qr_res using the $qr attribute and assign it to a variable named Q.
  4. We also extract the upper triangular matrix R from qr_res using the $qr attribute and assign it to a variable named R.
  5. We print both matrices Q and R to the console to see the results. This allows us to verify the decomposition.

R Program

mat <- matrix(c(1, 2, 3, 4, 5, 6, 7, 8), nrow = 4, byrow = TRUE)
qr_res <- qr(mat)
Q <- qr.Q(qr_res)
R <- qr.R(qr_res)
print('Orthogonal Matrix Q:')
print(Q)
print('Upper Triangular Matrix R:')
print(R)

Output

[1] "Orthogonal Matrix Q:"
           [,1]       [,2]
[1,] -0.1091089 -0.8295151
[2,] -0.3273268 -0.4391550
[3,] -0.5455447 -0.0487950
[4,] -0.7637626  0.3415650
[1] "Upper Triangular Matrix R:"
          [,1]        [,2]
[1,] -9.165151 -10.9108945
[2,]  0.000000  -0.9759001

Summary

In this tutorial, we learned How to Perform QR Decomposition in R language with well detailed examples.


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