Matrix Operations

The Matrix class implements the following matrix operations. The same rules as in mathematics apply.

All these operations return a new Matrix instance except stated otherwise.

Via unary operators

• Negation: -m
• Inverse: ~m

Via binary operators

• Equality comparison: m1 == m2
• Addition: m1 + m2
• Subtraction: m1 - m2
• Scalar multiplication: m * c or c * m
• Exponentiation (Repeated matrix multiplication): m ** c
• Matrix multiplication: m1 @ m2
• Division (by scalar): m / c
• Augmentation: m1 | m2

where m is a matrix and c is a real number.

The augmented assignment counterparts of these binary operators are also supported and perform the operations in-place i.e the matrix object remains unchanged.

Via explicit methods

• Transpose: This is implemented by two methods:
  • transposed() -> Returns a transposed copy.
  • transpose() -> Transposes the matrix in-place.
• Row reduction (all In-place): Implemented as instance methods:
  • to_upper_triangular() -> Reduces a square matrix to an upper-triangular matrix.
  • to_lower_triangular() -> Reduces a square matrix to a lower-triangular matrix.
  • to_row_echelon() -> Transforms the matrix to row echelon form.
  • to_reduced_row_echelon() -> Transforms the matrix to reduced row echelon form.
  • Note: The last two work on matrices of any shape.
• forward_eliminate() -> Performs forward elimination on an horizontal matrix, in-place.
• back_substitute() -> Performs back substitution on an horizontal matrix, in-place.
  • This operation requires that forward elimination must’ve been performed on the matrix.