Decompositions
This work builds atop incredible previous work.
Duality of Matrices
We often encounter matrices in two settings, either as a collection of measurements or as a linear operator. Decomposition tools can be regardless of what the underlying matrix represents. While decomposition is exactly the same for both settings, it is not immediately obvious how to reconcile what decomposition means in either case. For now, this is an open question for me.
Eigendecompositions
I often have to grapple with matrices in a linear dynamical system, particularly the discrete case \[ \mathbf{x}_{k+1} = \mathbf{A}\mathbf{x}_{k}, \]
where the matrix \(\mathbf{A} \in \mathbb{R}^{N \times N}\) is the transformation which brings the state \(\mathbf{x} \in \mathbb{R}^{N}\) from one time step \(k\) to the next \(k+1\).
If you shed all the context of “dynamics”, all that is going on is that the space is getting sheared. Maybe it
Citation
@online{aswani2025,
author = {Aswani, Nishant},
title = {Decompositions},
date = {2025-10-06},
url = {https://nishantaswani.com/articles/decompositions/},
langid = {en}
}