HESSENBERG Meaning and
Definition
-
Hessenberg is a term used in linear algebra to describe a specific type of matrix. A matrix is said to be Hessenberg if all of its elements below the subdiagonal are zero. In other words, a Hessenberg matrix can be visualized as having zeros in the lower left corner, with all other elements potentially non-zero.
More formally, a square matrix A of size n x n is Hessenberg if a(i, j) = 0 for all i > j + 1, where a(i, j) represents the element at the i-th row and j-th column of the matrix A. This defines a trapezoidal shape of non-zero elements along the subdiagonal, and zeros elsewhere in the matrix.
Hessenberg matrices have several important properties that make them useful in certain applications. For example, Hessenberg form is often employed in matrix algorithms as a way to simplify calculations, as most operations involving Hessenberg matrices have lower computational complexity compared to general matrices. Additionally, Hessenberg matrices are closely related to triangular matrices, which are even simpler to work with.
The name "Hessenberg" comes from the German mathematician Karl Hessenberg, who introduced this type of matrix in the early 20th century. Hessenberg matrices have since been extensively studied and found applications in various areas, including numerical analysis, ordinary differential equations, and control theory.
Etymology of HESSENBERG
The word "Hessenberg" is of German origin and derives from the name of the German state Hesse, which is called Hessen in German. It is specifically related to the mathematical concept known as Hessenberg matrix, which was first introduced by Karl Hessenberg, a German mathematician, in the early 20th century. The term "Hessenberg matrix" was coined in honor of his contributions in the field.
