Jul 19, 2025
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Special properties
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Eigens your vectors
Taking a slight tangent from classical optics, I decided to delve more into non-linear optics.
As someone who never had to use Gaussian form of Maxwell's equations, never used Tensors, and had never used Einsteins notation for summation; let me just say the algebra of the book proved to be a hard nut to crack!
What am I learning about? Anisotropy! Who knew that the direction at which you observe/propagate through a material such as a crystal plays a role in the material properties you will experience!!
I still don't believe you can determine eigenvalues and eigenvectors for an arbitrary 3x3 permittivity matrix (I need to use numerical examples to really see it) and that there exists rotational matricies that let us make the math all neat and proper.
But, slowly and with growing pains, I am continuing to slowly tread the waters of this fascinating topic!
"You must to choose your eigenvectors wisely, they aren't a one night stand that you can forget later"
Linear Algebra professor introducing eigenvectors
An eigenface (/ˈaɪɡənˌfeɪs/) is the name given to a set of eigenvectors when used in the computer vision problem of human face recognition. (via Wikipedia)
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A map between Math and Programming
Polynominals are eigenforms if seen as recursive functions.
Coefficients are eigenvectors because they change the scale/intensity.
Exponentiation scale value within a row.
The square root of negatives shift value to another axis.
I get Quaternions (you can also have negatives of exponentials i, j, k...).
Polynominals are symmetry groups with marked row.
The complex plane (and Quaternions, Octonions) are symmetry groups with a marked axis.
The independent variable is the input, the dependent variable is the output.
A symmetry group is a struct.
A symmetry group with pre-available shifts is an object.
Navigating a scalar shift in your independent values is running a program.
This week’s notes!
linear algebra, eigenvalues and eigenvectors are a fundamental concept with applications in various fields, from data compression to physics