Linear algebra topics
A Note on the Courant–Fischer–Weyl Theorem [Link]:
This is a brief note on the Courant--Fischer--Weyl Theorem, often referred to as the Courant--Fischer--Weyl min-max principle, a useful tool for calculating the k-th smallest eigenvalue of a Hermitian matrix by applying the min-max principle to the Rayleigh quotient of vectors in any k-dimensional subspace. This note provides the theorem statement and a proof.
A Note on the Cauchy Interlacing Theorem [Link]:
This brief note presents the Cauchy interlacing theorem, also known as the Poincaré separation theorem, which explores the relationships between the eigenvalues of a square matrix and those of its compression to a smaller square matrix. The proof of the Cauchy interlacing theorem leverages the Courant--Fischer--Weyl min-max principle, a powerful method for calculating the k-th smallest eigenvalue of a Hermitian matrix by applying the min-max principle to the Rayleigh quotient of vectors within any k-dimensional subspace. We assume the Courant--Fischer--Weyl min-max principle and demonstrate the proof of the Cauchy interlacing theorem based on this approach.
A Note on Vandermonde Matrices [Link]:
This note briefly introduces the definition of Vandermonde matrices and their determinants. Additionally, as an application, we present an example that demonstrates how Vandermonde matrices can be utilized to embed any finite d-dimensional abstract simplicial complex into the R^{2d+1} space.
A Note on Matrix Determinants and Alternating Maps [Link]:
The aim of this note is to introduce the connection between the matrix determinant and alternating maps. Specifically, assuming the definition of the matrix determinant using the Laplace expansion, we introduce alternating maps and the Leibniz formula for such maps. In particular, since the Laplace expansion-based definition of an n \times n matrix forms an alternating map, the uniqueness of the Leibniz formula provides an equivalent way to define the determinant of an n \times n matrix.
A Note on the Multiplication of Matrix Determinants [Link]:
The aim of this note is to show that for any n \times n matrices A and B with entries in a commutative ring with identity, the formula det(AB) = det(A)det(B) holds, using the Leibniz formula for determinants. In this note, we briefly recall the definition of alternating maps and the Leibniz formula over a commutative ring with identity and utilize this formula to obtain the desired formula.