In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
, an alternant matrix is a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
formed by applying a finite list of functions pointwise to a fixed column of inputs. An alternant determinant is the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
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
:
or, more compactly,
. (Some authors use the
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
of the above matrix.) Examples of alternant matrices include
Vandermonde matrices, for which
, and
Moore matrices In linear algebra, a Moore matrix, introduced by , is a matrix (mathematics), matrix defined over a finite field. When it is a square matrix its determinant is called a Moore determinant (this is unrelated to the Moore determinant of a quaternionic ...
, for which
.
Properties
* The alternant can be used to check the
linear independence
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
of the functions
in
function space. For example, let
and choose
. Then the alternant is the matrix