Inlinear algebra, analternant matrix is amatrix formed by applying a finite list of functions pointwise to a fixed column of inputs. Analternant determinant is thedeterminant of a square alternant matrix.

Generally, if${\displaystyle f_1,f_2,\dots ,f_n}$ are functions from a set${\displaystyle X}$ to a field${\displaystyle F}$, and${\displaystyle {\alpha }_1,{\alpha }_2,\dots ,{\alpha }_m\in X}$, then the alternant matrix has size${\displaystyle m\times n}$ and is defined by

or, more compactly,${\displaystyle M_{ij}=f_j({\alpha }_i)}$. (Some authors use thetranspose of the above matrix.) Examples of alternant matrices includeVandermonde matrices, for which${\displaystyle f_j(\alpha )={\alpha }^{j-1}}$, andMoore matrices, for which${\displaystyle f_j(\alpha )={\alpha }^{q^{j-1}}}$.

• The alternant can be used to check thelinear independence of the functions${\displaystyle f_1,f_2,\dots ,f_n}$ infunction space. For example, let${\displaystyle f_1(x)=\sin (x)}$,${\displaystyle f_2(x)=\cos (x)}$ and choose${\displaystyle {\alpha }_1=0,{\alpha }_2=\pi /2}$. Then the alternant is the matrix and the alternant determinant is${\displaystyle -1\ne 0}$. ThereforeM is invertible and the vectors${\displaystyle \{\sin (x),\cos (x)\}}$ form a basis for their spanning set: in particular,${\displaystyle \sin (x)}$ and${\displaystyle \cos (x)}$ are linearly independent.

• Linear dependence of the columns of an alternant doesnot imply that the functions are linearly dependent in function space. For example, let${\displaystyle f_1(x)=\sin (x)}$,${\displaystyle f_2=\cos (x)}$ and choose${\displaystyle {\alpha }_1=0,{\alpha }_2=\pi }$. Then the alternant is and the alternant determinant is 0, but we have already seen that${\displaystyle \sin (x)}$ and${\displaystyle \cos (x)}$ are linearly independent.

• Despite this, the alternant can be used to find a linear dependence if it is already known that one exists. For example, we know from the theory ofpartial fractions that there are real numbersA andB for which${\displaystyle \frac{A}{x+1}+\frac{B}{x+2}=\frac{1}{(x+1)(x+2)}}$. Choosing${\displaystyle f_1(x)=\frac{1}{x+1}}$,${\displaystyle f_2(x)=\frac{1}{x+2}}$,${\displaystyle f_3(x)=\frac{1}{(x+1)(x+2)}}$ and${\displaystyle ({\alpha }_1,{\alpha }_2,{\alpha }_3)=(1,2,3)}$, we obtain the alternant. Therefore,${\displaystyle (1,-1,-1)}$ is in thenullspace of the matrix: that is,${\displaystyle f_1-f_2-f_3=0}$. Moving${\displaystyle f_3}$ to the other side of the equation gives the partial fraction decomposition${\displaystyle A=1,B=-1}$.

• If${\displaystyle n=m}$ and${\displaystyle {\alpha }_i={\alpha }_j}$ for any${\displaystyle i\ne j}$, then the alternant determinant is zero (as a row is repeated).

• If${\displaystyle n=m}$ and the functions${\displaystyle f_j(x)}$ are all polynomials, then${\displaystyle ({\alpha }_j-{\alpha }_i)}$ divides the alternant determinant for all. In particular, ifV is aVandermonde matrix, then divides such polynomial alternant determinants. The ratio$\frac{detM}{detV}$ is therefore a polynomial in${\displaystyle {\alpha }_1,\dots ,{\alpha }_m}$ called thebialternant. TheSchur polynomial${\displaystyle s_{({\lambda }_1,\dots ,{\lambda }_n)}}$ is classically defined as the bialternant of the polynomials${\displaystyle f_j(x)=x^{{\lambda }_j}}$.

• Alternant matrices are used incoding theory in the construction ofalternant codes.

• List of matrices
• Wronskian

• Matrices (mathematics)
• Determinants