6.49.2  QR decomposition: qr

The QR decomposition of a square matrix A is A=QR, where Q is an orthogonal matrix (Q^TQ=I) and R is upper triangular. The qr command finds the QR decomposition of a matrix.

• qr takes one argument:
  A, a numeric square matrix.
• qr(A) returns a list [Q,R] with Q and R from the QR decomposition.

Examples.

• Input:

  A := [[3,5],[4,5]]:;

  qr(A)

  Output:

  ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 3 5 4 5 4 5 − 3 5 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

  ,

  ⎡ ⎢ ⎣

  5 7 0 1

  ⎤ ⎥ ⎦

• Input:
  qr([[1,2],[3,4]])
  Output:

  ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 √ 10 3 5  √ 10 5 3 √ 10 − 1 5  √ 10 5 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

  ,

  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

  √ 10 7 5   √ 10 0 √ 10 5

  ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦