16.14 Inverse

Contents

Description

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:

  1. Create a new matrix with data.
  2. Activate the matrix.
  3. Select Analysis: Mathematics: Inverse to open the minverse dialog box.

Dialog Options

Recalculate

Controls recalculation of analysis results

  • None
  • Auto
  • Manual

For more information, see: Recalculating Analysis Results

Input Matrix

Choose input matrix.

For help with range controls, see: Specifying Your Input Data

Output Matrix

Choose where to output the inverse matrix.

For help with the range controls, see: Output Results

Algorithm

For a square matrix of full rankA\!, the inverse matrix A^{-1}\! (aka the reciprocal matrix), will satisfy the relationship:

AA\!^{-1}=A\!^{-1}\!A=I\!

where I\! is the identity matrix.

The calculation of A^{-1}\! can be expressed as:

A^{-1}=\frac 1{|A|}A^{*}

where |A|\! is the determinant of matrix A\! and A^*\!is the adjoint.

A^*=\begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \end{bmatrix}

a_{ij}=\left( -1\right) ^{i+j}|A^{ij}|

where A^{ij}\! is the (n-1)\times (n-1) matrix by elimination of the i_{th}\!column and j_{th}\!row of A\!.

When matrices do not have inverses or determinants, a Moore-Penrose pseudoinverse will be computed. It exists for any (m,n)\! matrix.

Given an (m\times n) matrix A\!, A^+\!is the unique n\times m\! pseudoinverse matrix. If m>n\! and A has full rank, then the A^+\! satisfies the following:

A\!^{+}=(A^TA)\!^{-1}A\!^T

The computation is based on singular value decomposition (SVD) of the matrix A\! and any singular values within tolerance, are treated as zero. If the rank of A\! is not full, then the matrix will shrink to a smaller matrix.

References

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).