Hat Matrix Rank

Then the eigenvalues of Hare all either 0 or 1.

Hat matrix rank. 1A square matrix A is a projection if it is idempotent 2A projection A is orthogonal if it is also symmetric. For linear models the trace of the hat matrix is equal to the rank of X which is the number of independent parameters of the linear model. SinceHis an idempotent matrixXiXX1Xi is also idempotent.

It follows that the hat matrix His symmetric too. Sij weights Yjs contribution to mˆxi. Rank of a Matrix and Some Special Matrices The maximum number of its linearly independent columns or rows of a matrix is called the rank of a matrix.

From my understanding it should be 1 as the Rank of a 1-column-vector 1. The hat matrix projection matrix P in econometrics is symmetric idempotent and positive definite. The rank of a matrix cannot exceed the number of its rows or columns.

Where the hat matrix H Q 1Q0 1 is a projection matrix of rank trQ 1Q0 1 trQ 0 1Q 1 trI k k The diagonal matrix D in the singular value decomposition sect. For a given model with independent variables and a dependent variable the hat matrix is the projection matrix to. The rank is at least 1 except for a zero matrix a matrix made of all zeros whose rank is 0.

Using part a of Lemma 11 1. We can show that both H and I H are orthogonal projections. A symmetric idempotent matrix such as H is called a perpendicular projection matrix.

Just note that yˆ y e I My Hy 31 where H XX0X1X0 32 Greene calls this matrix P but he is alone. Prove that the elements of any column or row of H. I 1 n H ii m and i 1 n H ij 1.