## 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:

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.

2.

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

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

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:

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:

## 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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