DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3600 Linear Statistical Analysis
Chapter 2 Matrix and Distribution Theory
2 Matrix
A matrix is a rectangular array of numbers. A is an n × m matrix if it contains n rows and m columns of elements, where ai j represents the element of A at the i-th row and j-th column (i ∈ {1,...,n}, j ∈ {1,...,m}). Mathematically, we denote
When n = 1, A is a 1 × m matrix or a row vector. When m = 1, A is an n × 1 or a column vector. When m = n = 1, A is just a scalar.
2.1 Basic Matrix Operation
1. Matrix Addition:
Let A and B be both n×m matrices. C = A+B is an n×m matrix with elements C = (cij) where cij = ai j +bij. Similarly, D = A−B is also an n×m matrix with elements D = (dij) where dij = aij − bij.
2. Matrix Multiplication:
Suppose A is an n ×m matrix and B is an m ×p matrix. Then C = AB is an n ×p matrix with elements C = (cij) where
Note that AB ≠ B A. Moreover let k be a scalar, then D = k A is an n × m matrix with element D = (dij), dij = kaij.
2.2 Some Matrix Definition
1. Transpose:
Let A be an n × m matrix, an m × n matrix B = AT (or A') is called the transpose of A with elements B = (bij) where bij = aji.
It is also easy to see that (AT)T = A and (AB)T = BTAT.
2. Square Matrix:
A is a square matrix when n = m.
3. Symmetric:
A is symmetric when A is square and aij = aji for all i and j. For a symmetric matrix, A = AT.
4. Idempotent:
A is idempotent when A2 = A.
5. Identity Matrix:
A is an identity matrix, denoted by In or I , when aii = 1 for all i and aij = 0 for all i ≠ j.
6. Rank:
Let A be an n ×m matrix. The rank of A, denoted by rank(A), is defined as the maximum number of linearly independent column (or row) vectors of A. A is of full ranked if rank(A) = min(n, m). Moreover we have
rank(A) = rank(AT) ≤ min(n, m)
7. Inverse Matrix:
Let A be an n × n matrix. If there exists an n × n matrix B such that AB = I , then B is the inverse of A, and is denoted by B = A−1.
Some properties about the inverse of a matrix are listed below:
• AB = I → B A = I.
• If A−1 exists, then A is called a non-singular or invertible matrix.
• A square matrix A has an inverse if and only if it is of full ranked.
• The inverse is unique if it exists.
• (A-1)-1 = A.
• (AT)
−1 = (A−1)T.
• (AB)−1 = B−1 A−1. (Note: A and B are square matrices)
• (kA)−1 = k/1A-1 for some scalar k.
Inversion of 2 × 2 matrices:
Inversion of 3 × 3 matrices:
where det(A) = a (ek − fh) − b(kd − fg) + c(dh − eg).
8. Trace
For an n × n matrix A, the trace is defined as the sum of its diagonal elements
Moreover, we have tr(AB) = tr(B A).
9. Quadratic form.
Let W be a function of n variables Y1
,...,Yn where
where aij is a given set of numbers. The above form. can be written as W = YTAY where A is an n × n matrix with elements A = (aij), Y = (Y1
,...,Yn )T is a n × 1 column vector, and W is defined to be a quadratic form. in the n variables Yi
.
10. Nonnegative and positive definite
An n × n matrix A is said to be nonnegative definite (or positive semi-definite) if it is symmetric and YTAY ≥ 0 for any n × 1 vector Y .
It is positive definite if YTAY > 0 for any nonzero n × 1 vector Y .