16.14 Inverse

1 Description
2 Dialog Options
3 Algorithm
4 References

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:

Create a new matrix with data.
Activate the matrix.
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 rank $A\!$ , 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.