Semiparametric model

Template:Distinguish In linear algebra, an alternant matrix, is a matrix with a particular structure, in which successive columns have a particular function applied to their entries. An alternant determinant is the determinant of an alternant matrix. Such a matrix of size m × n matrix may be written out as

$M=\left[\begin{array}{cccc}{f}_1({\alpha }_1)& f_2({\alpha }_1)& \dots & f_n({\alpha }_1)\\ f_1({\alpha }_2)& f_2({\alpha }_2)& \dots & f_n({\alpha }_2)\\ f_1({\alpha }_3)& f_2({\alpha }_3)& \dots & f_n({\alpha }_3)\\ ⋮& ⋮& \ddots & ⋮\\ f_1({\alpha }_m)& f_2({\alpha }_m)& \dots & f_n({\alpha }_m)\\ \end{array}\right]$

or more succinctly

$M_{i,j}=f_j({\alpha }_i)$

for all indices i and j. (Some authors use the transpose of the above matrix.)

Examples of alternant matrices include Vandermonde matrices, for which $f_i(\alpha )={\alpha }^{i-1}$ and Moore matrices for which $f_i(\alpha )={\alpha }^{q^{i-1}}$.

If $n=m$ and the $f_j(x)$ functions are all polynomials we have some additional results: if ${\alpha }_i={\alpha }_j$ for any $i<j$ then the determinant of any alternant matrix is zero (as a row is then repeated), thus $({\alpha }_j-{\alpha }_i)$ divides the determinant for all $1\le i<j\le n$. As such, if we take

$V=\left[\begin{array}{cccc}1& {\alpha }_1& \dots & {\alpha }_1^{n-1}\\ 1& {\alpha }_2& \dots & {\alpha }_2^{n-1}\\ 1& {\alpha }_3& \dots & {\alpha }_3^{n-1}\\ ⋮& ⋮& \ddots & ⋮\\ 1& {\alpha }_n& \dots & {\alpha }_n^{n-1}\\ \end{array}\right]$

(a Vandermonde matrix) then $\prod _{i<j}({\alpha }_j-{\alpha }_i)=\det V$ divides such polynomial alternant determinants. The ratio $\frac{\det M}{\det V}$ is called a bialternant. In the case where each function $f_j(x)=x^{m_j}$, this forms the classical definition of the Schur polynomials.

Alternant matrices are used in coding theory in the construction of alternant codes.

See also

• List of matrices

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534