Econ310
Handout # 1 Some Useful Matrix Results
Notation: Capital letter s denote matrices. Small bold letters denote vectors. Small unbold letters denote
scalars. Unprimed small bold letters denote column vectors, primed small bold letters denote row vectors.
Things which you should know about matrices and vectors.
Matrix, column vector, row vector, square matrix, diagonal matrix, scalar matrix, identity
matrix, matrix equality, matrix addition and subtraction, scalar multiplication,
matrix multiplication,
A' = the transpose of A
A-1 = the inverse of A
(AB)' = B'A' assuming that A and B are conformable for multiplication.
(AB)-1 = B-1A-1 assuming both inverses exist.
(A')-1 = (A-1)'
DEF: a square matrix which has an inverse is said to be nonsingular
DEF: A is symmetric if A = A'
DEF: A is idempotent if AA = A
DEF: The trace of a square matrix is the sum of the diagonal elements.
DEF: Two vectors x and y are said to be orthogonal if
DEF: a set of vectors x1, x2, ... xn is said to be linearly dependent if there exists a set of scalars a1, a2, ...
an, not all zero, such the a1x1 + a2x2 + ... + anxn = 0. Otherwise the set of vectors is said to linearly
independent.
DEF: The rank of a matrix is the maximum number of linearly independent rows or columns. (Note:
Given an n x m matrix A, rank(A) minimum of n and m.)
Thm: For any matrix A, rank(A) = rank(A') = rank(A'A)
DEF: let A be a square matrix. A non-zero vector x is said to be an eigenvector of A if there exists a
scalar : such that A: = :x. : is called an eigenvalue of the matrix A.
DEF: x'Ax is called a quadratic form. (Note: any quadratic form can be expressed as a quadratic form
with a symmetric matrix.)
DEF: A is positive definite if x'Ax > 0 for every
DEF: A is positive semidefinite if for every
Thm: If A is symmetric and idempotent then
i) rank(A) = trace(A)
ii) A is positive semidefinite
