How to create a symmetric matrix using Python NumPy?

by | Oct 3, 2023 | Linear Algebra

Create Symmetric Matrix Using Python

What is a symmetric matrix?

A symmetric matrix is a matrix which is transpose of itself. In other words, if A is a square matrix such that AT = A, then A is a symmetric matrix.

Let’s look at an example. Let’s say A is a 3×3 square matrix with the following elements.

A=\begin{bmatrix} 6 & 7 & 8 \\ 7 & 6 & 6 \\ 8 & 6 & 2 \end{bmatrix}

At this point, if Ai,j is the element in the ith row and jth column of the matrix A and AT is the transpose of A, then we can say that:

A_{i,j}=A_{j,i}^{T}

So, the transpose of the matrix A will be:

A^T={\begin{bmatrix} 6 & 7 & 8 \\ 7 & 6 & 6 \\ 8 & 6 & 2 \end{bmatrix}}^T =\begin{bmatrix} 6 & 7 & 8 \\ 7 & 6 & 6 \\ 8 & 6 & 2 \end{bmatrix}

Hence, AT = A and A is a symmetric matrix.

How to create a symmetric matrix using Python NumPy?

If A is a square matrix, then (AT)T = A. So, we can say that, if A is any square matrix, then (A + AT)/2 is a symmetric matrix.

B=\frac{(A+A^T)}{2} \\ B^T=\frac{(A+A^T)^T}{2}=\frac{(A^T+A)}{2}=B

We can use the following Python code to create a symmetric matrix from a square matrix A…

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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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