📂Mathematical Statistics

Quadratic Form of Random Vector

Definition ¹

For a random vector $\mathbf{X} = \left( X_{1} , \cdots , X_{n} \right)$ and a symmetric matrix $A \in \mathbb{R}^{n \times n}$, $Q = \mathbf{X}^{T} A \mathbf{X}$ is called a quadratic form.

Explanation

Since the quadratic form $A = \left( a_{ij} \right)$ is a symmetric matrix, it can be expressed in several ways, as shown below, and is useful in many applications. $$\begin{align*} & Q \\ =& \mathbf{X}^{T} A \mathbf{X} \\ =& \sum_{i=1}^{n} \sum_{j=1}^{n} a_{ij} X_{i} X_{j} \\ =& \sum_{i=1}^{n} a_{ii} X_{i}^{2} + \sum_{i \ne j} a_{ij} X_{i} X_{j} \\ =& \sum_{i=1}^{n} a_{ii} X_{i}^{2} + 2 \sum_{i > j} a_{ij} X_{i} X_{j} \end{align*}$$

The theory of quadratic forms with respect to random vectors is particularly important in contexts such as the F-test.

See Also

• Quadratic forms in linear algebra