1. invertmatrix

Computes the inverse of a transformation matrix.

1.1. Syntax 

matrix1 matrix2 invertmatrix → matrix2

1.2. Stack Effects

Table 1. Before
Level Object

1

matrix1 (6-element array)

0

matrix2 (6-element array)
Table 2. After
Level Object

0

matrix2 (modified array)

1.3. Description

invertmatrix replaces the value of matrix2 with the inverse of matrix1 and pushes the modified matrix2 back on the operand stack.

The inverse matrix has the property that if matrix1 transforms a coordinate (x, y) to (x', y'), then matrix2 transforms (x', y') back to (x, y).

Mathematically: M₁ × M₂ = M₂ × M₁ = I (identity matrix)

If matrix1 is singular (determinant = 0), invertmatrix executes an [undefinedresult] error because the inverse does not exist.

1.4. PostScript Level

Level 1 and later

1.5. Examples

Inverting a translation
/forward matrix 100 200 translate def
/inverse matrix def
forward inverse invertmatrix
% inverse = [1 0 0 1 -100 -200]
% Translates by (-100, -200)
Inverting a scale
/scale2x matrix 2 2 scale def
/scale_half matrix def
scale2x scale_half invertmatrix
% scale_half = [0.5 0 0 0.5 0 0]
% Scales by 1/2
Inverting a rotation
/rot45 matrix 45 rotate def
/rot_neg45 matrix def
rot45 rot_neg45 invertmatrix
% rot_neg45 rotates by -45 degrees

1.6. Common Use Cases

1.6.1. Undoing Transformations 

% Apply transformation
/myTransform [2 0 0 2 100 100] def
myTransform concat

% Compute inverse
/myInverse matrix def
myTransform myInverse invertmatrix

% Undo transformation
myInverse concat
% Back to original coordinate system

1.6.2. Coordinate Space Conversion 

% Transform from device to user space
matrix currentmatrix /ctm exch def
/inverseCTM matrix def
ctm inverseCTM invertmatrix

% Now can transform device coordinates to user coordinates
% using inverseCTM with transform operator

1.6.3. Building Inverse Transformations 

% Create matched pair of transformations
/setupTransform {
  % Build forward transformation
  /forward matrix def
  forward 100 100 translate
  forward 2 2 scale
  forward 45 rotate

  % Build inverse
  /inverse matrix def
  forward inverse invertmatrix
} def

1.7. Common Pitfalls

Singular Matrices - Matrices with zero determinant cannot be inverted. 
% This matrix has determinant = 0
[2 4 1 2 0 0] invertmatrix  % Error: undefinedresult
% Because: (2×2) - (4×1) = 0
Numerical Precision - Very large or very small matrix values can cause precision loss. 
% Extreme values may lose precision
[1000000 0 0 1000000 0 0] invertmatrix
% Inverse has very small values (0.000001)
% May experience rounding errors
Order of Application - The inverse undoes transformations in reverse order. 
% Forward: translate then scale
matrix 100 100 translate 2 2 scale

% Inverse: scale⁻¹ then translate⁻¹
% (automatically computed by invertmatrix)
Check Validity Before Inverting - Verify matrix is non-singular: 
% Compute determinant
/checkMatrix {  % matrix -> bool
  dup 0 get exch dup 3 get mul  % a×d
  exch dup 1 get exch 2 get mul % b×c
  sub                            % a×d - b×c
  0 ne                           % determinant ≠ 0
} def

myMatrix checkMatrix {
  myMatrix inverse invertmatrix
} if
Use for Coordinate Conversion - Primary use is with itransform and idtransform: 
% Manual inverse transformation
matrix currentmatrix /ctm exch def
/inv matrix def
ctm inv invertmatrix
% Now can use inv with transform for inverse effect

1.8. Error Conditions

Error Condition

[rangecheck]

Either array has fewer than 6 elements

[stackunderflow]

Fewer than 2 operands on stack

[typecheck]

Operands are not arrays, or array elements are not all numbers

[undefinedresult]

matrix1 is singular (determinant = 0) and cannot be inverted

1.9. Implementation Notes

  • The inverse is computed using standard matrix inversion

  • Numerical stability depends on matrix condition number

  • Well-conditioned matrices invert accurately

  • Ill-conditioned matrices may experience precision loss

  • The determinant must be non-zero for inversion to succeed

1.10. Matrix Mathematics

For matrix M = [a b c d tx ty], the inverse M⁻¹ is:

det = a×d - b×c  (determinant)

If det ≠ 0:

M⁻¹ = [d/det  -b/det  -c/det  a/det
       (c×ty - d×tx)/det  (b×tx - a×ty)/det]

Properties of inverse:

  • M × M⁻¹ = M⁻¹ × M = I (identity)

  • (M⁻¹)⁻¹ = M

  • (M₁ × M₂)⁻¹ = M₂⁻¹ × M₁⁻¹

1.11. Special Cases

1.11.1. Identity Matrix Inverse 

[1 0 0 1 0 0] invertmatrix
% → [1 0 0 1 0 0]
% Identity is its own inverse

1.11.2. Translation Inverse 

[1 0 0 1 100 200] invertmatrix
% → [1 0 0 1 -100 -200]
% Inverse translates by negative amounts

1.11.3. Scale Inverse 

[2 0 0 3 0 0] invertmatrix
% → [0.5 0 0 0.333... 0 0]
% Inverse scales by reciprocals

1.11.4. Rotation Inverse 

% 45° rotation:
[0.707 0.707 -0.707 0.707 0 0] invertmatrix
% → [0.707 -0.707 0.707 0.707 0 0]
% -45° rotation (same as transpose for rotation)

1.12. Determinant

The determinant indicates if a matrix is invertible:

/determinant {  % matrix -> number
  dup 0 get exch dup 3 get mul  % a×d
  exch dup 1 get exch 2 get mul % b×c
  sub                            % a×d - b×c
} def

myMatrix determinant 0 eq {
  (Matrix is singular!) print
} {
  (Matrix is invertible) print
} ifelse

1.13. Performance Considerations

  • Matrix inversion is computationally intensive

  • More expensive than matrix multiplication

  • Avoid inverting repeatedly; compute once and reuse

  • Well-conditioned matrices invert faster

1.14. Relationship to Transform Operators

invertmatrix is used with inverse transformation operators:

% Forward transformation
/fwd matrix currentmatrix def
100 200 fwd transform  % User to device

% Inverse transformation
/inv matrix def
fwd inv invertmatrix
100 200 inv itransform % Device to user

1.15. See Also

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