Calculate the pseudoinverse of a matrix using Python

by | Oct 3, 2023 | Featured, Linear Algebra

Pseudoinverse Of A Matrix

What is the pseudoinverse of a matrix?

We know that if A is a square matrix with full rank, then A-1 is said to be the inverse of A if the following condition holds:

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

The pseudoinverse or the Moore-Penrose inverse of a matrix is a generalization of matrix inverse when the matrix is not invertible. And when the matrix is invertible, the pseudoinverse or Moore-Penrose inverse of the matrix is the same as the inverse of the matrix.

Let’s say A is an mxn matrix. A+ is said to be the pseudoinverse or Moore-Penrose inverse of A if the following conditions hold:

1.

AA^{+}A=A

2.

A^{+}AA^{+}=A^{+}

3. AA+ is Hermitian. So, the conjugate transpose of AA+ is AA+. In other words,

({AA^{+}})^*=AA^{+}

4. A+A is Hermitian. So, the conjugate transpose of A+A is A+A. In other words,

({A^{+}A})^*=A^{+}A

Please also note that if the rank of the matrix A is equal to its column rank, then A+ is said to be the left inverse of A if the following condition holds:

A^{+}A=I

On the other hand, if the rank of the matrix A is equal to its row rank, then A+ is said to be the right inverse of A if the following condition holds:

AA^{+}=I

How to calculate the pseudoinverse of a matrix using Python?

We can use the following Python code to calculate the pseudoinverse of a matrix…

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Amrita Mitra

Author

Ms. Amrita Mitra is an author, who has authored the books “Cryptography And Public Key Infrastructure“, “Web Application Vulnerabilities And Prevention“, “A Guide To Cyber Security” and “Phishing: Detection, Analysis And Prevention“. She is also the founder of Asigosec Technologies, the company that owns The Security Buddy.

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