Existance of solutions to set of quadratic equations

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SUMMARY

The discussion centers on the conditions for the existence of solutions to a set of quadratic equations involving two unknown vectors, x and y, constrained to real non-negative values with norms equal to one. The equations are represented as y^T A_i x = 0 for m known n x n matrices A_i. Key conclusions indicate that the minimal number of equations m and the linear independence of the matrices are critical factors influencing the existence of solutions. Specifically, if the matrices are linearly independent, there may be conditions under which no solutions exist.

PREREQUISITES
  • Understanding of vector norms and properties
  • Familiarity with quadratic forms and matrix operations
  • Knowledge of linear independence in the context of matrices
  • Basic concepts of real-valued matrices and their applications
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  • Research the implications of linear independence on solution sets in quadratic equations
  • Explore the properties of quadratic forms in relation to vector spaces
  • Study the role of matrix rank in determining the existence of solutions
  • Investigate specific cases of m equations and their impact on solution existence
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Mathematicians, researchers in applied mathematics, and anyone studying systems of equations involving quadratic forms and linear algebra.

Leo321
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We have two unknown vectors x,y with real non negative values.
We can assume ||x||=||y||=1.
There are m known nxn matrices with real values.
We have m equations of the form:
yTA1x=0
...
yTAmx=0

What are the conditions for the existence of solutions for x,y?
What is the minimal number m and the condition on the matrices(linear independence, etc) for there to be no solutions?

Thanks
 
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Any ideas?
 

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