mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a system of bilinear equations is a special sort of

system of polynomial equations A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations where the are polynomials in several variables, say , over some Field (mathematics), field . A ''solution'' of a polynomial system is a se ...

, where each equation equates a

bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...

with a constant (possibly zero). More precisely, given two sets of variables represented as

coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimension ...

s and ''y'', then each equation of the system can be written

y^TA_ix=g_i,

where, is an

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

whose value ranges from 1 to the number of equations, each

A_i

is a

matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

, and each

g_i

is a

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

. Systems of bilinear equations arise in many subjects including

engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...

biology Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...

, and

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

References

* Charles R. Johnson, Joshua A. Link 'Solution theory for complete bilinear systems of equations' - http://onlinelibrary.wiley.com/doi/10.1002/nla.676/abstract * Vinh, Le Anh 'On the solvability of systems of bilinear equations in finite fields' - https://arxiv.org/abs/0903.1156 * Yang Dian 'Solution theory for system of bilinear equations' - https://digitalarchive.wm.edu/handle/10288/13726 * Scott Cohen and Carlo Tomasi. 'Systems of bilinear equations'. Technical report, Stanford, CA, USA, 1997.- ftp://reports.stanford.edu/public_html/cstr/reports/cs/tr/97/1588/CS-TR-97-1588.pdf Equations {{algebra-stub

See also

References