How to create a skew symmetric matrix using Python NumPy?

by | Oct 3, 2023 | Linear Algebra

Skew-Symmetric Matrix

What is a skew symmetric matrix?

A skew symmetric matrix is a square matrix that is negative of its own transpose. In other words, if A is a square matrix, then A is a skew symmetric matrix if and only if the following condition holds true:

A=-A^T

Now, we know that if A is a square matrix, AT is the transpose of A and Ai,j and ATi,j are the elements in the ith row and jth column of A, and AT, respectively, then

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

So, for a skew symmetric matrix,

A=-A^T \\ A_{i,j}=A_{j,i}^{T}=-A_{j,i}

In other words, the element in the ith row and jth column of a skew symmetric matrix is negative of the element in the jth row and ith column. And all the diagonal elements are 0.

For example, the matrix A given below is a skew symmetric matrix.

A=\begin{bmatrix} 0 & 6 & 3 \\ -6 & 0 & 7 \\ -3 & -7 & 0 \end{bmatrix}

Here,

A^T={\begin{bmatrix} 0 & 6 & 3 \\ -6 & 0 & 7 \\ -3 & -7 & 0 \end{bmatrix}}^T \\ =\begin{bmatrix} 0 & -6 & -3 \\ 6 & 0 & -7 \\ 3 & 7 & 0 \end{bmatrix}=-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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