How to create an orthogonal matrix using Python NumPy?

by | Oct 3, 2023 | Linear Algebra

Create An Orthogonal Matrix

What is an orthogonal matrix?

A square matrix is said to be an orthogonal matrix if the following condition holds true:

AA^T=A^TA=I

Now, we know that, if A and B are two matrices such that the following condition holds true

AB=I \\ B=A^{-1}

then, B is said to be the inverse of the matrix A.

So, we can say that for an orthogonal matrix A:

AA^T=A^TA=I \\ A^T=A^{-1}

So, in other words, a square matrix A is an orthogonal matrix, if the transpose of the matrix A is equal to the inverse of the matrix A.

For example, letโ€™s say A is the following matrix:

A={\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}}

The transpose of A is:

A^T={\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}}

And,

AA^T={\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}} \times {\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}} \\ ={\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}}=I

Therefore,

AA^T=A^TA=I \\ A^T=A^{-1}

Hence, A is an orthogonal 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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