Is xTAx always non-zero for a real, symmetric, nonsingular matrix A?

Thread starter elimax
Start date Apr 29, 2015
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Form Quadratic Quadratic forms Symmetric matrix Zero

In summary, a quadratic form is a mathematical expression that includes variables raised to the second power and can also contain linear and constant terms. It is nonzero if at least one term has a nonzero coefficient and its graph does not intersect the x-axis. Knowing if a quadratic form is nonzero is important in understanding its behavior and solving equations. A quadratic form cannot be both zero and nonzero, and it is related to a quadratic equation as a general expression.

Apr 29, 2015

elimax

Basic question, I think, but I'm not sure. It is a step in a demonstration, so it would be nice if it were true.

True or false? Why? If A is a real, symmetric, nonsingular matrix, then xTAx≠0 for x≠0.

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Apr 29, 2015

Truecrimson

Since ##A## is real and symmetric, it is diagonalizable. So you may start by thinking about ##x^T A x## in a basis in which ##A## is diagonal.

FAQ: Is xTAx always non-zero for a real, symmetric, nonsingular matrix A?

What is a quadratic form?

A quadratic form is a mathematical expression that contains a combination of variables that are raised to the second power, such as x^2 or y^2. It can also include linear terms (variables raised to the first power) and constant terms. Quadratic forms are commonly used in algebra and calculus to model real-world situations and solve equations.

How do you determine if a quadratic form is nonzero?

A quadratic form is nonzero if it has at least one term with a nonzero coefficient. This means that the expression is not equal to zero for any input values of the variables. In other words, the graph of a nonzero quadratic form will not intersect the x-axis at any point.

What is the importance of knowing if a quadratic form is nonzero?

Knowing if a quadratic form is nonzero is important because it can help determine the behavior of the equation. A nonzero quadratic form will have a parabola-shaped graph, with either a minimum or maximum point depending on the sign of the leading coefficient. This information can be useful in solving equations and understanding real-world problems.

Can a quadratic form be both zero and nonzero?

No, a quadratic form cannot be both zero and nonzero. If a quadratic form is equal to zero, then all of its terms must be equal to zero. This means that the leading coefficient, as well as any other coefficients and variables, must be equal to zero. Therefore, the quadratic form would not be considered nonzero.

How is a quadratic form related to a quadratic equation?

A quadratic form is a general expression that can be used to represent any quadratic equation. A quadratic equation is a specific type of quadratic form that is set equal to zero. In other words, a quadratic equation is a quadratic form that is solved for a certain outcome, while a quadratic form is a more general representation of the equation.