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 f 1 , f 2 , … , f n {\displaystyle f_{1},f_{2},\dots ,f_{n}} are functions from a set X {\displaystyle X} to a field F {\displaystyle F} , and α 1 , α 2 , … , α m ∈ X {\displaystyle {\alpha _{1},\alpha _{2},\ldots ,\alpha _{m}}\in X} , then the alternant matrix has size m × n {\displaystyle m\times n} and is defined by
M = [ f 1 ( α 1 ) f 2 ( α 1 ) ⋯ f n ( α 1 ) f 1 ( α 2 ) f 2 ( α 2 ) ⋯ f n ( α 2 ) f 1 ( α 3 ) f 2 ( α 3 ) ⋯ f n ( α 3 ) ⋮ ⋮ ⋱ ⋮ f 1 ( α m ) f 2 ( α m ) ⋯ f n ( α m ) ] {\displaystyle M={\begin{bmatrix}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, M i j = f j ( α i ) {\displaystyle M_{ij}=f_{j}(\alpha _{i})} . (Some authors use the transpose of the above matrix.) Examples of alternant matrices include Vandermonde matrices, for which f j ( α ) = α j − 1 {\displaystyle f_{j}(\alpha )=\alpha ^{j-1}} , and Moore matrices, for which f j ( α ) = α q j − 1 {\displaystyle f_{j}(\alpha )=\alpha ^{q^{j-1}}} .
Properties
- The alternant can be used to check the linear independence of the functions f 1 , f 2 , … , f n {\displaystyle f_{1},f_{2},\dots ,f_{n}} in function space. For example, let f 1 ( x ) = sin ( x ) {\displaystyle f_{1}(x)=\sin(x)} , f 2 ( x ) = cos ( x ) {\displaystyle f_{2}(x)=\cos(x)} and choose α 1 = 0 , α 2 = π / 2 {\displaystyle \alpha _{1}=0,\alpha _{2}=\pi /2} . Then the alternant is the matrix [ 0 1 1 0 ] {\displaystyle \left[{\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}\right]} and the alternant determinant is − 1 ≠ 0 {\displaystyle -1\neq 0} . Therefore M is invertible and the vectors { sin ( x ) , cos ( x ) } {\displaystyle \{\sin(x),\cos(x)\}} form a basis for their spanning set: in particular, sin ( x ) {\displaystyle \sin(x)} and cos ( x ) {\displaystyle \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 f 1 ( x ) = sin ( x ) {\displaystyle f_{1}(x)=\sin(x)} , f 2 = cos ( x ) {\displaystyle f_{2}=\cos(x)} and choose α 1 = 0 , α 2 = π {\displaystyle \alpha _{1}=0,\alpha _{2}=\pi } . Then the alternant is [ 0 1 0 − 1 ] {\displaystyle \left[{\begin{smallmatrix}0&1\\0&-1\end{smallmatrix}}\right]} and the alternant determinant is 0, but we have already seen that sin ( x ) {\displaystyle \sin(x)} and cos ( x ) {\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 of partial fractions that there are real numbers A and B for which A x + 1 + B x + 2 = 1 ( x + 1 ) ( x + 2 ) {\displaystyle {\frac {A}{x+1}}+{\frac {B}{x+2}}={\frac {1}{(x+1)(x+2)}}} . Choosing f 1 ( x ) = 1 x + 1 {\displaystyle f_{1}(x)={\frac {1}{x+1}}} , f 2 ( x ) = 1 x + 2 {\displaystyle f_{2}(x)={\frac {1}{x+2}}} , f 3 ( x ) = 1 ( x + 1 ) ( x + 2 ) {\displaystyle f_{3}(x)={\frac {1}{(x+1)(x+2)}}} and ( α 1 , α 2 , α 3 ) = ( 1 , 2 , 3 ) {\displaystyle (\alpha _{1},\alpha _{2},\alpha _{3})=(1,2,3)} , we obtain the alternant [ 1 / 2 1 / 3 1 / 6 1 / 3 1 / 4 1 / 12 1 / 4 1 / 5 1 / 20 ] ∼ [ 1 0 1 0 1 − 1 0 0 0 ] {\displaystyle {\begin{bmatrix}1/2&1/3&1/6\\1/3&1/4&1/12\\1/4&1/5&1/20\end{bmatrix}}\sim {\begin{bmatrix}1&0&1\\0&1&-1\\0&0&0\end{bmatrix}}} . Therefore, ( 1 , − 1 , − 1 ) {\displaystyle (1,-1,-1)} is in the nullspace of the matrix: that is, f 1 − f 2 − f 3 = 0 {\displaystyle f_{1}-f_{2}-f_{3}=0} . Moving f 3 {\displaystyle f_{3}} to the other side of the equation gives the partial fraction decomposition A = 1 , B = − 1 {\displaystyle A=1,B=-1} .
- If n = m {\displaystyle n=m} and α i = α j {\displaystyle \alpha _{i}=\alpha _{j}} for any i ≠ j {\displaystyle i\neq j} , then the alternant determinant is zero (as a row is repeated).
- If n = m {\displaystyle n=m} and the functions f j ( x ) {\displaystyle f_{j}(x)} are all polynomials, then ( α j − α i ) {\displaystyle (\alpha _{j}-\alpha _{i})} divides the alternant determinant for all 1 ≤ i < j ≤ n {\displaystyle 1\leq i<j\leq n} . In particular, if V is a Vandermonde matrix, then ∏ i < j ( α j − α i ) = det V {\textstyle \prod _{i<j}(\alpha _{j}-\alpha _{i})=\det V} divides such polynomial alternant determinants. The ratio det M det V {\textstyle {\frac {\det M}{\det V}}} is therefore a polynomial in α 1 , … , α m {\displaystyle \alpha _{1},\ldots ,\alpha _{m}} called the bialternant. The Schur polynomial s ( λ 1 , … , λ n ) {\displaystyle s_{(\lambda _{1},\dots ,\lambda _{n})}} is classically defined as the bialternant of the polynomials f j ( x ) = x λ j {\displaystyle f_{j}(x)=x^{\lambda _{j}}} .
Applications
- Alternant matrices are used in coding theory in the construction of alternant codes.
See also
- Muir, Thomas (2003) [1960]. A treatise on the theory of determinants. Dover Publications. pp. 321–363. ISBN 978-0-486-49553-8. OCLC 52203124.
- Aitken, A.C. (1956). Determinants and Matrices (9th ed.). Oliver and Boyd Ltd. pp. 111–123. OCLC 271302373.
- Stanley, Richard P. (1999). Enumerative Combinatorics (2nd ed.). Cambridge University Press. pp. 334–342. doi:10.1017/CBO9781139058520. ISBN 978-1-107-01542-5. OCLC 897778191.