Eigenvalues and Eigenvectors

Definition

In linear algebra, a linear transformation refers to a mapping between two vector spaces that preserve operations of vector addition and scalar multiplication, such that $\mathrm{\mathcal{L}}:\mathcal{V}\to \mathcal{W}$, where $\left(\mathcal{V},\, \mathbb{F}\right)$ and $\left(\mathcal{W},\, \mathbb{F}\right)$ are vector spaces defined over a field $\mathbb{F}$. The commonly known linear transformations that can be expressed geometrically are rotation at a fixed point about an axis and scaling along the basis vectors. To understand rotation and scaling, they can be understood with respect to a three-dimensional (3D) object and the basis vectors, i.e., x, y, and zaxes in the Cartesian coordinate space. In computer graphics, an object is often represented using a surface mesh, discretized into triangles. The vertices of the triangles in the mesh can be rotated or scaled to rotate or scale the object, respectively. These vertices can be written as 3D column...