1 Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 6QF Matrix Solutions to Normal Equations

2 Direct (Kronecker) Products (Left) Direct Product Some Properties assuming all operations are valid

3 Solving the Normal Equations Single-Factor, Balanced Experiment y ij =  +  i + e ij i = 1,..., a; j = 1,..., r Matrix Formulation y = X  + e y = (y 11 y y 1r... y a1 y a2... y ar )’ Show

4 Properties of X’X Symmetric Same rank as X Has an inverse (X’X )-1 iff X has full column rank

5 Solving the Normal Equations Residuals Least Squares Solution: Solve the Normal Equations

6 Solving the Normal Equations Normal Equations Problem: X’X is Singular, has no inverse Show

7 Generalized Inverse: G or Definition Moore-Penrose Generalized Inverse AGA = A (i)AGA = A (ii)GAG = G (iii) AG is symmetric (iv) GA is symmetric not unique unique Some Properties if A has full row rank, G = A’(AA’) -1 if A has full column rank, G = (A’A) –1 A’ Common Notation A A

8 Solving the Normal Equations Every solution to the normal equations corresponds to a generalized inverse of X’X Every generalized inverse of X’X solves the normal equations Normal EquationsSolutions Theorem: For any, X’X X’ = X’ (X’X)

9 A Solution to the Normal Equations One Generalized InverseVerification

10 A Solution to the Normal Equations Corresponds to the solution to the normal equations with the constraint  = 0 imposed

11 Assignment Find another generalized inverse for X’X in a one-factor balanced experiment Verify that it is a generalized inverse Solve the normal equations using the generalized inverse Determine what constraint on the model parameters correspond to the generalized inverse