Calculate the inverse of a matrix using QR decomposition in Python

by | Oct 3, 2023 | Featured, Linear Algebra

Calculate Inverse Using QR Decomposition

In our previous article, we discussed QR decomposition of a matrix. We discussed that if Ais an mxn matrix with linearly independent columns, then using QR decomposition, we can express the matrix A as a product of two matrices Q and R, where Q is an orthogonal matrix and R is an upper triangular matrix. In other words,

A=QR

Q here is an orthogonal matrix. So, the following holds true for Q.

QQ^T=Q^TQ=I \\ Q^T=Q^{-1}

And R is an upper triangular matrix.

Now, if we invert both sides of the equation A = QR, then we get the following:

A=QR \\ A^{-1}=(QR)^{-1}=R^{-1}Q^{-1}

And, for an orthogonal matrix Q, the inverse of the matrix is equal to the transpose of the matrix.

Q^T=Q^{-1} \\ A^{-1}=R^{-1}Q^{-1}=R^{-1}Q^{T}

So, if we perform the QR decomposition of a square matrix A, and then calculate R-1QT, we will get the inverse of the matrix A.

Please note that R is an upper triangular matrix. So, many elements of the matrix R are zeros. So, it is easier to compute R-1 than to compute the inverse of the matrix A directly.

How to calculate the inverse of a matrix using QR decomposition in Python?

We can use the following Python code to calculate the inverse of a matrix A using QR decomposition…

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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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