Theory of Statistics - George Mason University

776 0 Statistical Mathematics
A matrix has an inverse iff it is square and of full rank.
Definition 0.3.6 (generalized inverse of a matrix)
For A ∈ IR n×m , a matrix B ∈ IR m×n such that ABA = A is called a
generalized inverse of A, and is written A − .
If A is nonsingular (square and full rank), then obviously A − = A −1 .
Definition 0.3.7 (pseudoinverse or Moore-Penrose inverse of a matrix)
For A ∈ IR n×m , the matrix B ∈ IR m×n such that ABA = A, BAB = B,
(AB) T = AB, and (BA) T = BA is called the pseudoinverse of A, and is
written A + .
Definition 0.3.8 (orthogonal matrix)
For A ∈ IR n×m , if A T A = Im, that is, if the columns are orthonormal and
m ≤ n, or AA T = In, that is, if the rows are orthonormal and n ≤ m, then A
is said to be orthogonal.
Definition 0.3.9 (quadratic forms)
For A ∈ IR n×n and x ∈ IR n , the scalar x T Ax is called a quadratic form.
Definition 0.3.10 (nonnegative definite matrix)
For A ∈ IR n×n and any x ∈ IR n , if x T Ax ≥ 0, then is said to be nonnegative
definite. We generally restrict the definition to symmetric matrices. This is
essentially without loss of generality because if a matrix is nonnegative definite,
then there is a similar symmetric matrix. (Two matrices are said to be
similar if they have exactly the same eigenvalues.) We write A 0 to denote
that A is nonnegative definite.
Definition 0.3.11 (positive definite matrix)
For A ∈ IR n×n and any x ∈ IR n , if x T Ax ≥ 0 and x T Ax = 0 implies x = 0,
then is said to be positive definite. As with nonnegative definite matrices,
we generally restrict the definition of positive definite matrices to symmetric
matrices. We write A ≻ 0 to denote that A is positive definite.
Definition 0.3.12 (eigenvalues and eigenvectors) If A ∈ IR n×n , v is an
n-vector (complex), and c is a scalar (complex), and Av = cv, then c is an
eigenvalue of A and v is an eigenvector of A associated with c.
Theory of Statistics c○2000–2013 James E. Gentle