Jekyll2025-09-15T02:39:46+00:00https://sarthakjshetty.github.io/feed.xmlSarthak’s CornerA place to (mostly) rambleSymmetric and Skew-Symmetric Matrices2024-02-26T04:28:09+00:002024-02-26T04:28:09+00:00https://sarthakjshetty.github.io/math/linear-algebra/2024/02/26/symmetric-skew-symmetricI’m starting to publish these notes, for two main reasons:

  1. I find myself revisiting these topics, especially when reading 3D vision papers, and Googling about them leads to weird SEO’d websites and secondly,
  2. I enjoy writing in LaTeX, and I’d like to get better at it; especially now that I no longer write longform LaTeX as part of coursework.

Context:

Symmetric matrices are matrices that have the property:

\[A = A^T \tag{1}\]

Skew-symmetric matrices on the other hand have a related but slightly different property:

\[A = -A^T \tag{2}\]

Property:

An interesting property that emerges as a consequence of this relationship, is that if a matrix \(A\) is skew-symmetric, then its inverse, \(A^{-1}\), is also skew-symmetric; i.e:

\[A^{-1} = -(A^{-1})^{T} \tag{3}\]

We can prove this from the above equation, and a familar property which is:

If \(A\) is invertible, then (I’ll show this proof in a future post):

\[(A^{-1})^{T} = (A^{T})^{-1} \tag{4}\]

Given this relationship, we can use Eq. \((4)\) in the RHS of Eq. \((3)\) to get:

\[-(A^{-1})^{T} = -(A^{T})^{-1}\tag{5}\]

And, since we established in Eq. \((1)\) that \(A\) is skew-symmetric already we have:

\[-(A^{-1})^{T} = -(A^{T})^{-1} = -(-A)^{-1} = A^{-1}\tag{6}\]

Therefore proving Eq. \((3)\)

I found this StackOverflow answer helpful [1] while recollecting these topics.

-Sarthak

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