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 \(Q_2\).