Determinant function

In linear algebra, a determinant function or determinant form is a special function that assigns a number to a sequence of vectors in a -dimensional vector space . ${\displaystyle n}$ ${\displaystyle n}$

definition

Let be a -dimensional vector space over a body . Then a function is called a determinant function if it fulfills the following conditions: ${\displaystyle V}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle f\colon V^{n}\rightarrow K}$

• ${\displaystyle f}$is multilinear , i.e. H. linear in each variable:

${\displaystyle \forall \,i\in \left\{1,\ldots ,n\right\},\;\forall \,a,b\in V\colon f\left(v_{1},\ldots ,v_{i-1},a+b,v_{i+1},\ldots ,v_{n}\right)=f\left(v_{1},\ldots ,v_{i-1},a,v_{i+1},\ldots ,v_{n}\right)+f\left(v_{1},\ldots ,v_{i-1},b,v_{i+1},\ldots ,v_{n}\right)}$ (Additivity)

${\displaystyle \forall \,i\in \left\{1,\ldots ,n\right\},\;\forall \,a\in V,\;\forall \,r\in K\colon f\left(v_{1},\ldots ,v_{i-1},r\cdot a,v_{i+1},\dots ,v_{n}\right)=r\cdot f\left(v_{1},\ldots ,v_{i-1},a,v_{i+1},\ldots ,v_{n}\right)}$ (Homogeneity)

• ${\displaystyle f}$is alternating :

${\displaystyle \left(\exists \,r,s\in \left\{1,\ldots ,n\right\},r\neq s\colon v_{r}=v_{s}\right)\Rightarrow f\left(v_{1},v_{2},\ldots ,v_{n}\right)=0}$

properties

• A determinant function is skew-symmetric , general applies to a permutation : where the Signum permutation called. ${\displaystyle \sigma }$ ${\displaystyle f\left(v_{\sigma (1)},v_{\sigma (2)},\dots ,v_{\sigma (n)}\right)=\mathrm {sgn} (\sigma )\cdot f\left(v_{1},v_{2},\dots ,v_{n}\right)}$ ${\displaystyle \mathrm {sgn} }$

• Are linearly dependent, then . For a non-trivial determinant function (i.e. ) the converse of this statement is also true. ${\displaystyle v_{1},v_{2},\dots ,v_{n}\in V}$ ${\displaystyle f(v_{1},v_{2},\dots ,v_{n})=0}$ ${\displaystyle f\not \equiv 0}$

• If there are two determinant functions and , then there is a such that . This means that apart from one normalization constant there is only one non-trivial determinant function, all other determinant functions can be obtained by multiplication with a constant. In fact, there is a non-trivial determinant function on every vector space. ${\displaystyle f,g:V^{n}\rightarrow K}$ ${\displaystyle f\not \equiv 0}$ ${\displaystyle a\in K}$ ${\displaystyle g(v_{1},v_{2},\dots ,v_{n})=a\cdot f(v_{1},v_{2},\dots ,v_{n})\;\forall \,v_{1},v_{2},\dots ,v_{n}\in V}$

Examples

• The null function is the so-called trivial determinant function.
• ${\displaystyle V=\mathbb {R} ^{n}}$, with the usual determinant as the determinant function.
• Determinant functions obtained from the previous example by multiplying the determinant by a constant.

literature

• H. Zieschang: Linear Algebra and Geometry . BG Teubner, Stuttgart 1997. ISBN 3-519-02230-3
• S. Bosch: Linear Algebra . Springer-Verlag, Münster 2008. ISBN 3-540-76437-2