quadratic form

USAGE SUMMARY

The phrase "quadratic form" is correct and usable in written English.
It is typically used in mathematics to refer to a polynomial of degree two in multiple variables. Example: "The quadratic form associated with the matrix can be used to analyze the curvature of the surface."

✓ Grammatically correct

Science

Academia

Human-verified examples from authoritative sources

Exact Expressions

58 human-written examples

Variance, covariance [? as ?] a quadratic form.

Massachusetts Institute of Technology

A simple calculation with these matrices resulted in a binary quadratic form.

The Guardian

Just this one calculation, actually it is quadratic form.

Massachusetts Institute of Technology

This quadratic form turns out to be unconditionally positive definite.

International Journal of Solids and Structures

And then you get the quadratic form, w transpose times this matrix w.

Massachusetts Institute of Technology

Dolciani's dissertation, "On the Representation of Integers by Quadratic Form", was supervised by B.W. Jones.

Cornell University

Monte Carlo simulation, the way I'll show you is based on a quadratic form.

Massachusetts Institute of Technology

The model for fat loss was in linear and quadratic form, whereas the model for moisture loss was in full quadratic form.

Meat Science

The arguments of the function undergo transformations that maintain a certain invariance of the function due to the quadratic form of the integrand.

Nature

The PFC is then designed through a linear quadratic form that makes the controller optimal.

Chemometrics and Intelligent Laboratory Systems

Human-verified similar examples from authoritative sources

Similar Expressions

1 human-written examples

The quadratic form and ellipsoidal isosurfaces as an intuitive way of understanding symmetric matrices.

University of California, Berkeley

Expert writing Tips

Best practice

When discussing mathematical models, be precise in using "quadratic form" to denote a specific type of polynomial expression. Avoid using it loosely to refer to any quadratic equation.

Common error

Avoid assuming that any equation involving squared terms is a "quadratic form". The term specifically refers to a homogeneous polynomial of degree two, not just any equation with a quadratic term.

Linguistic Context

The phrase "quadratic form" functions as a technical term in mathematics and related fields. Ludwig's examples illustrate its usage in academic and scientific writing to describe a specific algebraic expression.

Expression frequency: Very common

Frequent in

Science

76%

Academia

18%

Wiki

3%

Less common in

News & Media

1%

Formal & Business

1%

Encyclopedias

1%

Ludwig's WRAP-UP

The term "quadratic form" is a mathematical term denoting a homogeneous polynomial of degree two. Ludwig AI confirms its grammatical correctness and high frequency in scientific and academic contexts. It is essential to use this term precisely, distinguishing it from general quadratic equations. Related terms include "quadratic expression" and "quadratic polynomial", offering alternative ways to describe similar mathematical concepts. When using this term, ensure clarity and precision to maintain the integrity of mathematical discussions.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

quadratic expression

Focuses on the polynomial aspect rather than the specific algebraic structure.

quadratic polynomial

Highlights the polynomial nature and degree of the expression.

second-degree form

Emphasizes the degree of the polynomial in the expression.

binary quadratic form

Specifies that the quadratic form involves two variables.

positive definite quadratic form

Indicates that the quadratic form always yields positive values for nonzero inputs.

negative definite quadratic form

Indicates that the quadratic form always yields negative values for nonzero inputs.

indefinite quadratic form

Describes a quadratic form that can yield both positive and negative values.

homogeneous polynomial of degree two

Highlights the polynomial aspect and the degree of homogeneity.

sum of squares

Refers to a specific type of quadratic form where all terms are squared.

quadratic functional

Extends the concept to functionals, which are functions of functions.

FAQs

How is "quadratic form" used in a mathematical context?

In mathematics, a "quadratic form" is a homogeneous polynomial of degree two in a number of variables. It's often represented in matrix notation and used in various areas such as linear algebra and optimization.

What are some examples of "quadratic forms"?

Examples of "quadratic forms" include expressions like x² + 2xy + y², 3x² - y², and 5x² + 2yz + z². These expressions consist of terms where the total degree of each term is two.

How does a "quadratic form" differ from a general quadratic equation?

A "quadratic form" is a homogeneous polynomial of degree two, meaning every term has a total degree of two. A general quadratic equation, however, can include terms of degree one or zero. Thus, the expression "x²+2x+1" is a quadratic equation but not a quadratic form.

What is the significance of the discriminant in a "quadratic form"?

The discriminant of a "quadratic form" provides information about the nature of the solutions or roots of the associated quadratic equation. It helps determine whether the form is positive definite, negative definite, or indefinite.