chi square matrix form

n c = µ Aµ σ2 – If n c = 0, we simply say that it is a chi-square distribution – A symmetric matrix A (i e A = A), which is also idempotent (i e A2 = A) is called a projection matrix (a) Show that I (i e the identity matrix) is the matrix of Q1 and that J (i e an n by n matrix of 1's) is the matrix of Q2

For a random vector X and a fixed symmetric matrix A, X AX is called a quadratic function or quadratic form of X A necessary and sufficient condition for X AX is chi-square distributed is A2 = A, in which case the degrees of freedom of the chi-square distribution is the rank of A and the noncentrality parameter µ Aµ

An identity matrix is a square matrix of order p whose diagonal elements are unity (ones) and all The quadratic form 'X AX and the matrix A of the form is called If U has a noncentral Chi-square distribution with k degrees of freedom and 

12 juil 2004 · with mean vector µ and variance covariance matrix Σ (denoted y ∼ N(µ, Σ) ) if 3 both quadratic forms are distributed as chi-square variables

Review of Some Facts about Matrices, Quadratic Forms, and the has a noncentral chi-square distribution with k degrees of freedom and noncentrality param-

is,ˆY = PY for some projection matrix P Also, sinceˆY is in the space spanned by Definition: The non-central chi-squared distribution with n degrees of freedom 

and variance-covariance matrix Σ, and let χ2 m(λ) be the noncentral chi-square distribution with m degrees of freedom and noncentrality parameter λ The two

µ and a positive semidefinite matrix Σ, Y ∼ Nn(µ,Σ) if: In linear model theory, test statistics arise from sums of squares (special cases of quadratic forms) with χ2 4 23 Definition: The non-central chi-squared distribution with n degrees of