My current understanding of the eigensystem came about after my PhD. I took bra-ket notation — a tool for complex systems defying visualization — and used bra-kets for real-valued geometry in 3D space. I noticed that bra-ket notation made it easier to identify projectors and and ortho-projectors, especially in vector derivatives. This gave me geometric insight into the nature of vector derivatives and unlocked a coordinate-free understanding of linear algebra. I finally understood the determinant as the product of eigenvalues. Bra-ket notation empowered me to see results (the forest) that were previously hidden by coordinates (the trees).

I used bra-kets to understand the covariance and the Kalman filter. Finding no similar reference, and finding myself re-deriving important results, I wrote up a paper (in my free time) to describe my incremental journey. Most recently I added some similar explanations and personal references about least-squares and signal processing.