Question

Let \(Q_{1}\) and \(Q_{2}\) be two nonnegative quadratic forms in the observations of a random sample from a distribution which is \(N\left(0, \sigma^{2}\right) .\) Show that another quadratic form \(Q\) is independent of \(Q_{1}+Q_{2}\) if and only if \(Q\) is independent of each of \(Q_{1}\) and \(Q_{2}\) Hint: \(\quad\) Consider the orthogonal transformation that diagonalizes the matrix of \(Q_{1}+Q_{2}\). After this transformation, what are the forms of the matrices \(Q, Q_{1}\) and \(Q_{2}\) if \(Q\) and \(Q_{1}+Q_{2}\) are independent?

Short answer

The quadratic form \(Q\) is independent of each of \(Q_1\) and \(Q_2\) if and only if each linear form from \(Q\) is orthogonal to every linear form from \(Q_1\) and \(Q_2\). This conclusion is made clearer through the use of orthogonal transformation to diagonalize the matrix of \(Q_1+Q_2\), thus simplifying the task.

Step-by-step solution

Understand the independent quadratic forms

Quadratic forms are independent if and only if every linear form from one is orthogonal to every linear form from the second. Orthogonality in this case will refer to zero covariance. Therefore, to show that the quadratic form \(Q\) is independent of each \(Q_1\) and \(Q_2\), it must be shown that each linear form from \(Q\) is orthogonal to every linear form from \(Q_1\) and \(Q_2\).

Diagonalization Using Orthogonal transformation

Diagonalization using orthogonal transformation allows for simplification of algebraic structures by converting them into a canonical form. Apply an orthogonal transformation that will diagonalize the matrix of \(Q_1+Q_2\). The new forms of the matrices \(Q\), \(Q_1\) and \(Q_2\) will depend on whether they are independent of \(Q_1+Q_2\). The transformed forms will make clearer the interrelations among \(Q\), \(Q_1\) and \(Q_2\).

Conclude on the independence from the transformed forms

After the transformation, observe the transformed forms of the matrices. If the quadratic form \(Q\) is independent of \(Q_1+Q_2\), then the transformed form of \(Q\) should be orthogonal to linear forms obtained from \(Q_1\) and \(Q_2\). Now, it can be concluded that \(Q\) is independent of each of \(Q_1\) and \(Q_2\).

Key concepts

independence of quadratic forms

Quadratic forms involve expressions where a set of variables are multiplied and squared, offering a convenient way to represent multi-dimensional data. In statistics, understanding their independence is quite fundamental, especially when dealing with random variables. Two quadratic forms are considered independent if, loosely speaking, the influence of one has no impact on the other. To formalize this, think about linear forms - these are simply linear combinations of variables. For quadratic forms to be independent, every linear form from one quadratic form must be orthogonal to every linear form from the other.

• Orthogonality here implies that there is zero covariance between the linear forms.
• This means changes in one linear form will not affect the other.
• Thus, if quadratic form \(Q\) is independent of \(Q_1+Q_2\), it must individually be independent of \(Q_1\) and \(Q_2\).

To fully grasp the independence of quadratic forms, one must consider the influence correlations and covariances have on the quadratic expressions.

orthogonal transformation

An orthogonal transformation is a powerful tool often used to simplify complex mathematical structures. When you perform an orthogonal transformation, you essentially rotate the space of your variables, simplifying the relationships between them. This is particularly useful in transforming and handling quadratic forms. By using an orthogonal transformation, we can convert a matrix into a diagonal form, making its properties easier to analyze.

• Diagonalization simplifies interactions between variables because the matrix now contains nonzero elements only on its diagonal.
• This transformation ensures that each variable influences itself alone, without crossing effects.
• In the context of quadratic forms \(Q_1\), \(Q_2\), and a third form \(Q\), diagonalization helps clarify if \(Q\) remains independent from the sum \(Q_1+Q_2\).

Applying this transformation allows you to determine if \(Q\) and \(Q_1 + Q_2\) are orthogonal, thus confirming their independence.

diagonalization

Diagonalization is a process used to convert a matrix into its simplest form, where all non-diagonal elements are zero. This drastically reduces mathematical complexity when examining data as everything becomes about managing a series of unconnected parts. In terms of quadratic forms, diagonalization helps in distinguishing individual components contributing to the form, enabling easier understanding and verification of relationships such as independence.

• The matrix, once diagonalized, highlights the distinct contributions of each variable.
• For quadratic forms, this can reveal if a form \(Q\) interacts with the components of another form \(Q_1 + Q_2\), or remains independent.
• This process is especially useful in statistical computations, ensuring accuracy in multivariate analysis.

Thus, when we use diagonalization to explore the independence of quadratic forms, we gain clear insights into how forms relate to each other and validate theoretical conclusions drawn from data.