Method on @sym: [Q, R] = qr (A) ¶

Method on @sym: [Q, R] = qr (A, 0) ¶

Method on @sym: R = qr (A) ¶

Symbolic QR factorization of a matrix.

Example:

A = sym([1 1; 1 0]);

[Q, R] = qr (A)
  ⇒ Q = (sym 2×2 matrix)

      ⎡√2   √2 ⎤
      ⎢──   ── ⎥
      ⎢2    2  ⎥
      ⎢        ⎥
      ⎢√2  -√2 ⎥
      ⎢──  ────⎥
      ⎣2    2  ⎦

  ⇒ R = (sym 2×2 matrix)

      ⎡    √2⎤
      ⎢√2  ──⎥
      ⎢    2 ⎥
      ⎢      ⎥
      ⎢    √2⎥
      ⎢0   ──⎥
      ⎣    2 ⎦

Passing an extra argument of 0 gives an “economy-sized” factorization. This is currently the default behaviour even without the extra argument. In fact the result may even more minimal than the double precision ‘qr’; when the input has low rank, a rectangular rank-revealing result is returned:

[Q, R] = qr (sym ([6 2; 6 2]))
  ⇒ Q = (sym 2×1 matrix)
      ⎡√2⎤
      ⎢──⎥
      ⎢2 ⎥
      ⎢  ⎥
      ⎢√2⎥
      ⎢──⎥
      ⎣2 ⎦
  ⇒ R = (sym) [6⋅√2  2⋅√2]  (1×2 matrix)

We have one column in Q because the original matrix had rank one:

Q*R
  ⇒ ans = (sym 2×2 matrix)
      ⎡6  2⎤
      ⎢    ⎥
      ⎣6  2⎦

But what of the extreme case of a zero matrix?

[Q, R] = qr (sym (zeros (2, 2)))
  ⇒ Q = (sym) []  (empty 2×0 matrix)
  ⇒ R = (sym) []  (empty 0×2 matrix)

Q*R
  ⇒ ans = (sym 2×2 matrix)
      ⎡0  0⎤
      ⎢    ⎥
      ⎣0  0⎦

Magic? Not really but don’t let anyone tell you the dimensions of empty matrices are unimportant!

See also: qr, @@sym/lu.

Package: symbolic