Quadratic Form | Generated by AI

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In linear algebra, a quadratic form is a function $Q: \mathbb{R}^n \rightarrow \mathbb{R}$ that can be expressed as:

$Q(\mathbf{x}) = \mathbf{x}^T A \mathbf{x}$

where:

In simpler terms:

A quadratic form is a homogeneous polynomial of degree two in $n$ variables. “Homogeneous” means that every term in the polynomial has a total degree of two.

For example, in two variables ($n=2$), a quadratic form looks like: $Q(x_1, x_2) = a x_1^2 + b x_1 x_2 + c x_2^2$

This can be written in matrix form as: $Q(x_1, x_2) = \begin{bmatrix} x_1 & x_2 \end{bmatrix} \begin{bmatrix} a & b/2 \ b/2 & c \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix}$

Notice that the off-diagonal entries of the matrix $A$ are half the coefficient of the cross-term ($x_1 x_2$). We use a symmetric matrix $A$ because for any matrix $B$, $\mathbf{x}^T B \mathbf{x} = \mathbf{x}^T \left( \frac{B + B^T}{2} \right) \mathbf{x}$, and $\frac{B + B^T}{2}$ is always a symmetric matrix. Using the symmetric form simplifies many properties and theorems related to quadratic forms.

Key Aspects of Quadratic Forms:

Applications of Quadratic Forms:

Quadratic forms appear in various areas of mathematics, physics, statistics, and engineering, including:

In summary, a quadratic form is a fundamental concept in linear algebra that provides a way to express and analyze homogeneous quadratic polynomials using the language of vectors and matrices. Its matrix representation and classification based on definiteness are crucial for understanding its properties and applications.


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