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Resources on Linear Algebra

Matrix Generalized Inverse

By Kardi Teknomo, PhD.
LinearAlgebra

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Let us recalled how we define the matrix inverse. A matrix inverse generalized inverseis defined as a matrix that produces identity matrix when we multiply with the original matrixgeneralized inverse that is we definegeneralized inverse. Matrix inverse exists only for square matrices.

Real world data is not always square. Furthermore, real world data is not always consistent and might contain many repetitions. To deal with real world data, generalized inverse for rectangular matrix is needed.

Generalized inverse matrix is defined asgeneralized inverse. Notice that the usual matrix inverse is covered by this definition becausegeneralized inverse. We use term “generalized” inverse for a general rectangular matrix and to distinguish from inverse matrix that is for a square matrix. Generalized inverse is also called pseudo inverse.

Unfortunately there are many types of generalized inverse. Most of generalized inverse are not unique. Some of generalized inverse are reflexive generalized inverseand some are not reflexive. In this linear algebra tutorial, we will only discuss a few of them that often used in many practical applications.


For example:

  • Reflexive generalized inverse is defined as
    generalized inverse
    generalized inverse
  • Minimum norm generalized inverse such that generalized left inversewill minimizes generalized left inverseis defined as
    generalized inverse
    generalized inverse
    generalized left inverse
  • Generalized inverse that produces least square solution that will minimize residual or error generalized right inverseis defined as
    generalized inverse
    generalized right inverse

 

Left Inverse and Right Inverse

The usual matrix inverse is defined as two-sided inverse generalized inversebecause we can multiply the inverse matrix from the left or from the right of matrix generalized inverseand we still get the identity matrix. This property is only true for a square matrixgeneralized inverse.

For a rectangular matrixgeneralized inverse, we may have generalized left inverse or left inverse for short when we multiply the inverse from the left to get identity matrixgeneralized left inverse. Similarly, we may have generalized right inverse or right inverse for short when we multiply the inverse from the right to get identity matrixgeneralized right inverse. In general, left inverse is not equal to the right inverse.

Generalized inverse of a rectangular matrix is connected with solving of system linear equationslinear system. The solution to normal equation is Moore Penrose Generalized Inversewhich is equal toGeneralized Inverse. The term Generalized Left Inverseis often called as generalized left inverse. Still another pseudo inverse can also be obtained by multiplying the transpose matrix from the right and this is called generalized right inverseGeneralized Right Inverse.

 

Moore-Penrose Inverse

It is possible to obtain unique generalized matrix. To distinguish unique generalized inverse from other non-unique generalized inversesMoore Penrose Generalized Inverse, we use symbolMoore Penrose Generalized Inverse. The unique generalized inverse is called Moore Penrose inverse. It is defined using 4 conditions:
1.       Moore Penrose Generalized Inverse
2.       Moore Penrose Generalized Inverse
3.       Moore Penrose Generalized Inverse
4.       Moore Penrose Generalized Inverse

The first condition Moore Penrose Generalized Inverseis the definition of generalized inverse. Together with the first condition, the second condition indicates the generalized inverse is reflexive (generalized inverse). Together with the first condition, the third condition indicates the generalized inverse is the least square solution that minimizes the norm of errorgeneralized right inverse. The four conditions above make the generalized inverse unique.

Moore-Penrose inverse can be obtained through Singular Value Decomposition (SVD): Moore Penrose Generalized Inversesuch thatMoore Penrose Generalized Inverse.

Given a rectangular matrix, the interactive program below produces Moore-Penrose generalized inverse. The Random Example button will generate random matrix. Additionally, the output also include several matrices to let you test the 4 Moore Penrose conditions (Moore Penrose Generalized Inverse,Moore Penrose Generalized Inverse,Moore Penrose Generalized Inverse,Moore Penrose Generalized Inverse) and generalized left inverse or right inverse or matrix inverse depending on the input matrix size. The results can be shown in either rational format or decimal format.


Report in rational format

Properties

Some important properties of matrix generalized inverse are

  • The transpose of the left inverse of generalized inverse is the right inverseMoore Penrose Generalized Inverse. Similarly, the transpose of the right inverse of generalized inverse  is the left inverseMoore Penrose Generalized Inverse.
  • A matrix Moore Penrose Generalized Inversehas a left inverseMoore Penrose Generalized Inverse if and only if its rank equals its number of columns and the number of rows is more than the number of columnMoore Penrose Generalized Inverse.  In this caseMoore Penrose Generalized Inverse.
  • A matrix Moore Penrose Generalized Inversehas a right inverse Moore Penrose Generalized Inverseif and only if its rank equals its number of rows and the number of rows is less than the number of columns Moore Penrose Generalized Inverse . In this caseMoore Penrose Generalized Inverse.
  • Moore Penrose inverse is equal to left inverse Moore Penrose Generalized Inversewhen Moore Penrose Generalized Inverseand equal to right inverse Moore Penrose Generalized InversewhenMoore Penrose Generalized Inverse. Moore Penrose inverse is equal to matrix inverse Moore Penrose Generalized InversewhenMoore Penrose Generalized Inverse.

See also: Solving System of Linear Equations, Singular Value Decomposition, matrix Inverse, matrix transpose

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This tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\

 

 
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