⌹ — Matrix inverse / Matrix divide

Keys: Alt-q u. Ranks: 2 monadic, ∞ 2 dyadic

⌹Y gives an inverse, or a full-column-rank least-squares pseudoinverse. Units reciprocate.

⌹2ₓ                ⍝ 1r2
⌹[1 0⋄0 2]ₓ     ⍝ [1ₓ 0ₓ⋄0ₓ 1r2]

On an array of rank 3 or more, ⌹Y inverts each matrix, and X⌹Y divides X by each matrix.

⌹2 2 2⍴[1 0 0 1 2 0 0 4]ₓ   ⍝ 2 2 2⍴[1 0 0 1 1r2 0 0 1r4]ₓ

X⌹Y solves Y+.×Z = X, using least squares when overdetermined.

[3 5 7]ₓ⌹[1 1⋄1 2⋄1 3]ₓ      ⍝ [1 2]ₓ

All-exact inputs use rational elimination. Approximate real/complex inputs use pivoted LU for square systems and SVD for rectangular least squares.

Inverse

⌹⁻¹Y is ⌹Y.

⌹⁻¹[1 0⋄0 2]ₓ   ⍝ [1 0⋄0 1r2]ₓ

Errors

• DOMAIN: singular or underdetermined system, non-finite input