QR decomposition: qr

### 5.49.2 QR decomposition: qr

The QR decomposition of a square matrix A is A=QR, where Q is an orthogonal matrix (QTQ=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
  | ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦