alternant matrix

• The alternant can be used to check the linear independence of the functions $f\_1,\; f\_2,\; \backslash dots,\; f\_n$ in function space. For example, let {{nowrap|$f\_1(x)\; =\; \backslash sin(x)$,}} $f\_2(x)\; =\; \backslash cos(x)$ and choose $\backslash alpha\_1\; =\; 0,\; \backslash alpha\_2\; =\; \backslash pi/2$. Then the alternant is the matrix and the alternant determinant is {{nowrap|$-1\; \backslash neq\; 0$.}} Therefore M is invertible and the vectors $\backslash \{\backslash sin(x),\; \backslash cos(x)\backslash \}$ form a basis for their spanning set: in particular, $\backslash sin(x)$ and $\backslash cos(x)$ are linearly independent.

• Linear dependence of the columns of an alternant does not imply that the functions are linearly dependent in function space. For example, let {{nowrap|$f\_1(x)\; =\; \backslash sin(x)$,}} $f\_2\; =\; \backslash cos(x)$ and choose $\backslash alpha\_1\; =\; 0,\; \backslash alpha\_2\; =\; \backslash pi$. Then the alternant is and the alternant determinant is 0, but we have already seen that $\backslash sin(x)$ and $\backslash cos(x)$ 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 of partial fractions that there are real numbers A and B for which {{nowrap|$\backslash frac\{A\}\{x+1\}\; +\; \backslash frac\{B\}\{x+2\}\; =\; \backslash frac\{1\}\{(x+1)(x+2)\}$.}} Choosing {{nowrap|$f\_1(x)\; =\; \backslash frac\{1\}\{x+1\}$,}} {{nowrap|$f\_2(x)\; =\; \backslash frac\{1\}\{x+2\}$,}} $f\_3(x)\; =\; \backslash frac\{1\}\{(x+1)(x+2)\}$ and {{nowrap|$(\backslash alpha\_1,\backslash alpha\_2,\backslash alpha\_3)\; =\; (1,2,3)$,}} we obtain the alternant . Therefore, $(1,-1,-1)$ is in the nullspace of the matrix: that is, $f\_1\; -\; f\_2\; -\; f\_3\; =\; 0$. Moving $f\_3$ to the other side of the equation gives the partial fraction decomposition {{nowrap|$A\; =\; 1,\; B\; =\; -1$.}}

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

• If $n\; =\; m$ and the functions $f\_j(x)$ are all polynomials, then $(\backslash alpha\_j\; -\; \backslash alpha\_i)$ divides the alternant determinant for all {{nowrap|.}} In particular, if V is a Vandermonde matrix, then divides such polynomial alternant determinants. The ratio $\backslash frac\{\backslash det\; M\}\{\backslash det\; V\}$ is therefore a polynomial in $\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_m$ called the bialternant. The Schur polynomial $s\_\{(\backslash lambda\_1,\; \backslash dots,\; \backslash lambda\_n)\}$ is classically defined as the bialternant of the polynomials $f\_j(x)\; =\; x^\{\backslash lambda\_j\}$.

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

• List of matrices
• Wronskian

{{refbegin}}

• {{cite book |first=Thomas |last=Muir | authorlink=Thomas Muir (mathematician) | title=A treatise on the theory of determinants |orig-date=1960 | publisher=Dover Publications | pages=321–363 |isbn=978-0-486-49553-8 |oclc=52203124 |date=2003}}
• {{cite book |first=A.C. |last=Aitken | authorlink=Alexander Aitken | title=Determinants and Matrices | date=1956 | publisher=Oliver and Boyd Ltd | pages=111–123 |edition=9th |oclc=271302373}}
• {{cite book |first=Richard P. |last=Stanley | authorlink=Richard P. Stanley | title=Enumerative Combinatorics | date=1999 | publisher=Cambridge University Press | pages=334–342 |edition=2nd |doi=10.1017/CBO9781139058520 |isbn=978-1-107-01542-5 |oclc=897778191}}

{{refend}}

{{Matrix classes}}

Category:Matrices (mathematics)

Category:Determinants