Pseudo Inverse of a Matrix

To understand the Pseudo inverse of a matrix, first learn matrices, their inverse, and then the pseudo inverse.

A matrix is an array of values arranged in rows and columns. The inverse of a matrix is a matrix that gives the identity of a matrix when multiplied by itself.

\(A \cdot A^{-1} = I
\)

\(
A^{-1} \text{ is the inverse of matrix A}

\)

\(
\text{and I is an identity matrix}

\)

However, not all matrices have an inverse. The inverse of a matrix is only possible

• if a matrix is a square matrix (num of rows = num of columns)
• determinant of a matrix is not zero (non-singular matrix)

However, not all matrices are square matrices and have non-zero determinants. That's where Pseudo Inverse comes into the picture.

What is the Pseudo Inverse of a Matrix (Moore–Penrose Inverse)?

Pseudo Inverse, also known as Moore-Penrose Inverse, is a generalization of the inverse of a matrix that is not invertible. So now, we can solve the system of linear equations even if the matrix is not invertible.

It was introduced by E. H. Moore in 1920 and independently by Arne Bjerhammar in 1951. It was later generalized by Roger Penrose in 1955. So it got the name Moore-Penrose Inverse.

I didn't find any reference as to why it doesn't have the name of Bjerhammar in it.

Read more about the History of Moore-Penrose inverse

Algorithm of Moore Penrose Pseudo Inverse

Let M be a matrix of size j x k. Then the pseudo Inverse of Matrix M is denoted by M^+ and is defined as:

\(
M^{+} = (M^T \cdot M)^{-1} \cdot M^T


\)

Where

\(M^T\)

is the transpose of matrix M

\(M^-1\)

is the inverse of matrix M

How to calculate the Pseudo Inverse of a Matrix?

To calculate the pseudo inverse, either use the pseudo inverse calculator or follow the below steps:

1. Calculate the inverse of the Product of the transpose of matrix F and matrix F (F^T * F)^-1
2. Multiply it again with the transpose of matrix F (F^T)
3. The result is the pseudo inverse of matrix F

\(
F^{+} = (F^T \cdot F)^{-1} \cdot F^T


\)

Properties of Pseudo Inverse

The properties of Pseudo Inverse of a matrix are:

• The pseudo inverse of a matrix is unique
• Inverse of Matrix is same as Pseudo Inverse if inverse exists

Pseudo inverse can be used to find to solve the system of linear equation even if the matrix has no inverse. Not only that, it is also widely used in:

• Least Squares Problems
• Singular Value Decomposition
• Linear Regression
• Image Processing
• Control Theory
• Robotics
• Machine Learning

FAQ

What is the use of Pseudo Inverse of a matrix?

Pseudo Inverse can be used to solve linear equations even if the matrix is not invertible.

What is a pseudo inverse of a matrix?

Pseudo Inverse, also known as Moore-Penrose Inverse, is a near-to inverse of a matrix that is not invertible.

What is the difference between inverse and pseudo-inverse matrices?

The difference between inverse and pseudo-inverse matrix is:

• The inverse is Limited to a square matrix only and requires the matrix to be non-degenerate. It represents a one-to-one reversal of matrix transformation.

• Pseudo Inverse applies to any size matrix and can be computed even for degenerate matrix. It acts like an inverse of a matrix.

What is a pseudo inverse of a diagonal matrix?

The pseudo inverse of a diagonal matrix is the matrix, where all the elements except zero elements are replaced by their reciprocal.

Suppose if A is a diagonal matrix, then A⁺ is a matrix where all the non-zero elements are replaced by their reciprocal.

i.e; a_ii = 1/a_ii if a_ii != 0

What is a pseudo inverse of a zero matrix?

The pseudo inverse of a zero matrix is its transpose.

Is pseudo inverse same as the invertible?

If the matrix is invertible, then pseudo inverse and inverse of matrix are same. But if the matrix is not invertible, then pseudo inverse is a near-to inverse of matrix.

Is pseudo inverse of a zero matrix is its transpose?

Yes, pseudo inverse of a zero matrix is its transpose.