## Archive of posts with tag “linear algebra”

### Some understanding of Grassmann numbers out of intuition

I was briefly introduced to Grassmann numbers when I studied quantum field theory. I then had the natural question of how we can formally define them. In this article, I went with my intuition and tried to answer this question.

- Categories: math
- Tags: quantum field theory, linear algebra, abstract algebra

### Kinetic energy, momentum, and angular momentum of rigid bodies

In this article, we will find that the inertia matrix naturally appears when we calculate the kinetic energy $T$ or the angular momentum $\mathbf M$ of a rigid body. Then, we introduce the concept of principal inertia $\mathbf J_{\mathrm{pri}}$. We also study how the inertia matrix changes under translations and rotations and how those transformations may lead to conclusions that can help us simplify the calculation of inertia matrices.

- Categories: physics
- Tags: rigid body, linear algebra, classical mechanics, from zhihu

### Solving linear homogeneous ODE with constant coefficients

By using power series, we can prove that the problem of solving linear homogeneous ODE with constant coefficients can be reduced to the problem of solving a polynomial with those coefficients. This article illustrates this point in detail, but it uses a very awful notation…

- Categories: math
- Tags: calculus, linear algebra, combinatorics, ode, long paper, from zhihu

### Algebraic structure of chemicals

Regarding balancing chemical equations, actually we are trying to find a non-trivial linear combination of some chemicals to get zero. The interesting thing is that the coefficients can only be integers ($\mathbb Z$), which is not a field, so the algebraic structure of chemicals is not linear space. They actually live in a free $\mathbb Z$-module. See how I formalize this idea in mathematical language.

- Categories: chemistry
- Tags: linear algebra, abstract algebra

### Hyperellipsoids in barycentric coordinates

In this article, I introduce the barycentric coordinates: it is an elegant way to represent geometric shapes related to a simplex. By using it, given a simplex, we can construct a hyperellipsoid with the properties: its surface passes every vertex of the simplex, and its tangent hyperplane at each vertex is parallel to the hyperplane containing all other vertices.

- Categories: math
- Tags: linear algebra, long paper