The minverse X-Function generates an inverse matrix by dividing the adjoint by its determinant. When matrices do not have inverses or determinants, a Moore-Penrose pseudoinverse is computed.
For user-interface access to this function:
- Create a new matrix with data.
- Activate the matrix.
- Select Analysis: Mathematics: Inverse to open the minverse dialog box.
For a square matrix of full rank, the inverse matrix (aka the reciprocal matrix), will satisfy the relationship:
where is the identity matrix.
The calculation of can be expressed as:
where is the determinant of matrix and is the adjoint.
where is the matrix by elimination of the column and row of .
When matrices do not have inverses or determinants, a Moore-Penrose pseudoinverse will be computed. It exists for any matrix.
Given an matrix , is the unique pseudoinverse matrix. If and A has full rank, then the satisfies the following:
The computation is based on singular value decomposition (SVD) of the matrix and any singular values within tolerance, are treated as zero. If the rank of is not full, then the matrix will shrink to a smaller matrix.
- 1. E. H. Moore: On the reciprocal of the general algebraic matrix. Bulletin of the American Mathematical Society 26, 394-395 (1920).
- 2. Roger Penrose: A generalized inverse for matrices. Proceedings of the Cambridge Philosophical Society 51, 406-413 (1955).