Alternative matrix

Not to be confused with the Alternate Matrix .

Content

Alternate matrix ^[1] ^[2] ( Eng. Alternant matrix ) - in linear algebra a matrix of a special kind of dimension $m\times n$ ${\ displaystyle m \ times n}$ $m \ times n$ defined by $m$ ${\ displaystyle m}$ $m$ elements $\alpha _{1},\alpha _{2},\dots \alpha _{m}$ ${\ displaystyle \ alpha _ {1}, \ alpha _ {2}, \ dots \ alpha _ {m}}$ $\ alpha _ {1}, \ alpha _ {2}, \ dots \ alpha _ {m}$ and $n$ ${\ displaystyle n}$ $n$ functions $f_{1},f_{2},\dots f_{n}$ ${\ displaystyle f_ {1}, f_ {2}, \ dots f_ {n}}$ $f_ {1}, f_ {2}, \ dots f_ {n}$ so that each element of the matrix $M_{i,j}=f_{j}(\alpha _{i})$ ${\ displaystyle M_ {i, j} = f_ {j} (\ alpha _ {i})}$ $M _ {{i, j}} = f_ {j} (\ alpha _ {i})$ ^[3] or, in expanded form:

M={\begin{bmatrix}f_{1}(\alpha _{1})&f_{2}(\alpha _{1})&\dots &f_{n}(\alpha _{1})\\f_{1}(\alpha _{2})&f_{2}(\alpha _{2})&\dots &f_{n}(\alpha _{2})\\f_{1}(\alpha _{3})&f_{2}(\alpha _{3})&\dots &f_{n}(\alpha _{3})\\\vdots &\vdots &\ddots &\vdots \\f_{1}(\alpha _{m})&f_{2}(\alpha _{m})&\dots &f_{n}(\alpha _{m})\\\end{bmatrix}}

{\ displaystyle M = {\ begin {bmatrix} f_ {1} (\ alpha _ {1}) & f_ {2} (\ alpha _ {1}) & \ dots & f_ {n} (\ alpha _ {1}) \\ f_ {1} (\ alpha _ {2}) & f_ {2} (\ alpha _ {2}) & \ dots & f_ {n} (\ alpha _ {2}) \\ f_ {1} (\ alpha _ {3}) & f_ {2} (\ alpha _ {3}) & \ dots & f_ {n} (\ alpha _ {3}) \\\ vdots & \ vdots & \ ddots & \ vdots \\ f_ {1 } (\ alpha _ {m}) & f_ {2} (\ alpha _ {m}) & \ dots & f_ {n} (\ alpha _ {m}) \\\ end {bmatrix}}}

M = {\ begin {bmatrix} f_ {1} (\ alpha _ {1}) & f_ {2} (\ alpha _ {1}) & \ dots & f_ {n} (\ alpha _ {1}) \\ f_ {1} (\ alpha _ {2}) & f_ {2} (\ alpha _ {2}) & \ dots & f_ {n} (\ alpha _ {2}) \\ f_ {1} (\ alpha _ {3 }) & f_ {2} (\ alpha _ {3}) & \ dots & f_ {n} (\ alpha _ {3}) \\\ vdots & \ vdots & \ ddots & \ vdots \\ f_ {1} (\ alpha _ {m}) & f_ {2} (\ alpha _ {m}) & \ dots & f_ {n} (\ alpha _ {m}) \\\ end {bmatrix}}

Sometimes an alternative matrix is defined in transposed form .

Examples and use of alternative matrices

A common and frequent special case of an alternative matrix is the Vandermonde matrix . An alternative matrix takes this form when $f_{i}(\alpha )=\alpha ^{i-1}$ ${\ displaystyle f_ {i} (\ alpha) = \ alpha ^ {i-1}}$ . (Some authors refer to the Vandermond matrix as the alternative ^[4] ^[5] .) A rarer special case of the alternative matrix is the Moore matrix , wherein $f_{i}(\alpha )=\alpha ^{q^{i-1}}$ ${\ displaystyle f_ {i} (\ alpha) = \ alpha ^ {q ^ {i-1}}}$ .

More generally, alternative matrices are used in coding theory .

Alternative Matrix Properties

If the original alternative matrix is square and if all functions $f_{j}(x)$ ${\ displaystyle f_ {j} (x)}$ polynomial then provided $\alpha _{i}=\alpha _{j}$ ${\ displaystyle \ alpha _ {i} = \ alpha _ {j}}$ for all $i<j$ ${\ displaystyle i <j}$ the determinant of the alternative matrix is zero, and thus $(\alpha _{j}-\alpha _{i})$ ${\ displaystyle (\ alpha _ {j} - \ alpha _ {i})}$ is a divisor of the determinant of such an alternative matrix for any $i, j$ ${\ displaystyle i, j}$ satisfying the condition $1\leq i<j\leq n$ ${\ displaystyle 1 \ leq i <j \ leq n}$ . Therefore, the determinant of Vandermonde

V={\begin{bmatrix}1&\alpha _{1}&\dots &\alpha _{1}^{n-1}\\1&\alpha _{2}&\dots &\alpha _{2}^{n-1}\\1&\alpha _{3}&\dots &\alpha _{3}^{n-1}\\\vdots &\vdots &\ddots &\vdots \\1&\alpha _{n}&\dots &\alpha _{n}^{n-1}\\\end{bmatrix}}

{\ displaystyle V = {\ begin {bmatrix} 1 & \ alpha _ {1} & \ dots & \ alpha _ {1} ^ {n-1} \\ 1 & \ alpha _ {2} & \ dots & \ alpha _ {2} ^ {n-1} \\ 1 & \ alpha _ {3} & \ dots & \ alpha _ {3} ^ {n-1} \\\ vdots & \ vdots & \ ddots & \ vdots \\ 1 & \ alpha _ {n} & \ dots & \ alpha _ {n} ^ {n-1} \\\ end {bmatrix}}}

equal $\prod _{i<j}(\alpha _{j}-\alpha _{i})$ ${\ displaystyle \ prod _ {i <j} (\ alpha _ {j} - \ alpha _ {i})}$ is also a divisor of the determinants of such alternative matrices. Attitude ${\frac {\det M}{\det V}}$ ${\ displaystyle {\ frac {\ det M} {\ det V}}}$ bears the special name “ bialternant ”.

We also note that in the case when $f_{j}(x)=x^{m_{j}}$ ${\ displaystyle f_ {j} (x) = x ^ {m_ {j}}}$ , we get the classical definition of Schur polynomials .

Literature

AC Aitken. Determinants and Matrices. - 9th edition. - Edinburgh: Oliver and Boyd Ltd, 1956. - S. 111-123. - 144 p.
Richard P. Stanley. Enumerative Combinatorics. - Cambridge University Press, 1999. - T. 2. - S. 334–342. - ISBN 0521560691 .
Thomas Muir . A treatise on the theory of determinants. - Mineola, NY: Dover Publications, 2003 .-- S. 321-363. - 766 p. - ISBN 0486495531 .

Notes

↑ Alternant matrix (neopr.) . Academic.ru. Date of treatment November 17, 2012. Archived January 9, 2013.
↑ Alternant matrix (neopr.) . Multitran.ru. Date of treatment November 17, 2012.
↑ AC Aitken. Determinants and Matrices. - 9th edition. - Edinburgh: Oliver and Boyd Ltd, 1956 .-- S. 112 .-- 144 p.
↑ Hrishikesh D. Vinod. Hands-on matrix algebra using R: active and motivated learning with applications. - Singapore: World Scientific, 2011 .-- S. 290 .-- 329 p. - ISBN 9814313688 .
↑ Marvin Marcus, Henryk Minc. A survey of matrix theory and matrix inequalities. - New York: Dover, 1992 .-- S. 15 .-- 180 p. - ISBN 048667102X .

[1] Alternant matrix (neopr.) . Academic.ru. Date of treatment November 17, 2012. Archived January 9, 2013.

[2] Alternant matrix (neopr.) . Multitran.ru. Date of treatment November 17, 2012.

[3] AC Aitken. Determinants and Matrices. - 9th edition. - Edinburgh: Oliver and Boyd Ltd, 1956 .-- S. 112 .-- 144 p.

[4] Hrishikesh D. Vinod. Hands-on matrix algebra using R: active and motivated learning with applications. - Singapore: World Scientific, 2011 .-- S. 290 .-- 329 p. - ISBN 9814313688 .

[5] Marvin Marcus, Henryk Minc. A survey of matrix theory and matrix inequalities. - New York: Dover, 1992 .-- S. 15 .-- 180 p. - ISBN 048667102X .