Existance of solutions to set of quadratic equations

In summary, quadratic equations are algebraic expressions in the form of ax^2 + bx + c = 0, where a, b, and c are constants and x is a variable. They can have one or two solutions, which represent the values of x that make the equation true. To determine if a set of quadratic equations has solutions, you can use the quadratic formula. If the discriminant is greater than or equal to 0, then the set of equations has solutions. A set of quadratic equations can only have two solutions at most, as stated by the fundamental theorem of algebra. Some quadratic equations do not have solutions, as the discriminant can be negative, resulting in no real solutions. The number of solutions to a set
  • #1
Leo321
38
0
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
 
Last edited:
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  • #2
Any ideas?
 

1. What are quadratic equations?

Quadratic equations are algebraic expressions in the form of ax^2 + bx + c = 0, where a, b, and c are constants and x is a variable. They can have one or two solutions, which represent the values of x that make the equation true.

2. How do you determine if a set of quadratic equations has solutions?

To determine if a set of quadratic equations has solutions, you can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. If the discriminant (b^2 - 4ac) is greater than or equal to 0, then the set of equations has solutions.

3. Can a set of quadratic equations have more than two solutions?

No, a set of quadratic equations can only have two solutions at most. This is because a quadratic equation is a second-degree polynomial, and the fundamental theorem of algebra states that a polynomial of degree n can have at most n solutions.

4. Do all quadratic equations have solutions?

No, some quadratic equations do not have solutions. This occurs when the discriminant (b^2 - 4ac) is negative, meaning the solutions would involve imaginary numbers. Therefore, the set of equations would have no real solutions.

5. How does the number of solutions to a set of quadratic equations relate to its graph?

The number of solutions to a set of quadratic equations is equal to the number of times its graph intersects with the x-axis. If the graph intersects twice, then there are two solutions. If the graph only intersects once, then there is only one solution. If the graph does not intersect at all, then there are no solutions.

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