mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 . A ''solution'' of a polynomial system is a set of values for the ...

, 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 linear i ...

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

coordinates 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-dimensiona ...

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 (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

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

A_i

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 ...

, 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

. Systems of bilinear equations arise in many subjects including

engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...

biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...

, and

statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

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 {{algebra-stub Equations

See also

References