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The document presents a series of theorems related to quadratic forms and their properties in the context of multivariate normal distributions. It discusses the distribution of linear transformations of normal variables, chi-square distributions derived from quadratic forms, and conditions for independence between different quadratic forms. Key results include theorems on the behavior of quadratic forms under various matrix conditions, particularly focusing on idempotent matrices and their implications in statistical theory.

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What is the quadratic form in normal variables?

well-known theorems below establish sufficient and necessary conditions for the independence and distributions of quadratic forms in normal variates. Theorem 1. Let x Nk(,), 0, and A and B be k k real sym- metric matrices. Then xAx and xBx are independently distributed if and only if AB = 0.

What is the quadratic form of a variable?

A quadratic form in the variables x1,x2,,xn is a linear combination of monomials of degree 2. If A is any invertible matrix then hx,yi = Ax Ay is an inner product and hx,xi is a quadratic form. In general, if B is a symmetric matrix then xT Bx is a quadratic form and any quadratic form is of this kind.

What is the quadratic form in normal random variables?

Let A be such a matrix, and let X Nn(𝝁, 𝚺) with 𝚺 0. The scalar random vari- able (hereafter abbreviated r.v.) Y = XAX is referred to as a quadratic form (in normal variables).

What is the Craig theorem for quadratic forms?

2.6. Quadratic Form Theorem 6 (Craigs Theorem). Theorem 6. If y N(, ) where is positive definite, then q1 = yAy and q2 = yBy are independently distributed if AB = 0.

What is a normal quadratic equation?

Usually, the quadratic equation is represented in the form of ax2+bx+c=0, where x is the variable and a,b,c are the real numbers a 0. Here, a and b are the coefficients of x2 and x, respectively. So, basically, a quadratic equation is a polynomial whose highest degree is 2.