alternant matrix
{{Distinguish|alternating sign matrix}}
In linear algebra, an alternant matrix is a matrix formed by applying a finite list of functions pointwise to a fixed column of inputs. An alternant determinant is the determinant of a square alternant matrix.
Generally, if are functions from a set to a field , and , then the alternant matrix has size and is defined by
:
f_1(\alpha_1) & f_2(\alpha_1) & \cdots & f_n(\alpha_1)\\
f_1(\alpha_2) & f_2(\alpha_2) & \cdots & f_n(\alpha_2)\\
f_1(\alpha_3) & f_2(\alpha_3) & \cdots & f_n(\alpha_3)\\
\vdots & \vdots & \ddots &\vdots \\
f_1(\alpha_m) & f_2(\alpha_m) & \cdots & f_n(\alpha_m)\\
\end{bmatrix}
or, more compactly, . (Some authors use the transpose of the above matrix.) Examples of alternant matrices include Vandermonde matrices, for which , and Moore matrices, for which .
Properties
- The alternant can be used to check the linear independence of the functions in function space. For example, let {{nowrap|,}} and choose . Then the alternant is the matrix and the alternant determinant is {{nowrap|.}} Therefore M is invertible and the vectors form a basis for their spanning set: in particular, and 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|,}} and choose . Then the alternant is and the alternant determinant is 0, but we have already seen that and 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|.}} Choosing {{nowrap|,}} {{nowrap|,}} and {{nowrap|,}} we obtain the alternant . Therefore, is in the nullspace of the matrix: that is, . Moving to the other side of the equation gives the partial fraction decomposition {{nowrap|.}}
- If and for any {{nowrap|,}} then the alternant determinant is zero (as a row is repeated).
- If and the functions are all polynomials, then divides the alternant determinant for all {{nowrap|.}} In particular, if V is a Vandermonde matrix, then divides such polynomial alternant determinants. The ratio is therefore a polynomial in called the bialternant. The Schur polynomial is classically defined as the bialternant of the polynomials .
Applications
- Alternant matrices are used in coding theory in the construction of alternant codes.
See also
References
{{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}}